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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| i p c / 36 2 ■) . ELEMENTS OP . Infinitesimal Calculus. BT JOSEPH BAYMA, S. J., Prqfemr of Ma£h£iMU^ in Santa CUxra OoUege, 8,J,, Santa Clara, Califomia. San Fkancisoo : A. WALDTEUFEL, 787 MARKET STREET. \ COPTRIGHT, 1889, By a. WALDTEUFBL. w "^ ^ > > PREFACE The present treatise, as its title points out, has been based on the principles of the infinitesimal -^ method. I think that this metliod, besides being ^ the simplest of all for teaching Calculus, as all nf admit, is also the most consistent and philoso- \ phical. Some modern writers, indeed, have tried r< to discredit it ; but, when we examine their reason- ^ ings, we soon discover that they have done so be- cause they have failed to grasp the true nature of infinitesimal quantities. This I have endeavored to show in mv introduction to this work, where the reader will find not only what I believe to be the exact notion of the infinitesimal, but also a hint at the grounds (elsewhere developed) upon wliich the infinitesimal method rests its claim to be pre- ferred to its fashionable rival, the method of limits. This work being intended for young men who are supposed to devote a considerable part of their time to the study of mental and of natural philoso- phy, it has been necessary to limit its developments by giving less prominence to the analytical than to the practical portion of it. But, while anxious not PREFACE. to overtax our young people, I have nevertheless collected and condensed all that seemed to be of practical use in this branch of study; and, though I have made it a point to be concise, I have constantly endeavored to make all things as plain as the subject matter permitted. I hope that the average student will need only a mode- rate effort to understand the object and the pro- cesses of differentiation and of integration as laid down in this treatise ; though, as to integration, his success will often depend also on his endurance of analytical work. To assist him in the performance of this task, I have developed a sufficient number of geometrical and of mechanical problems, to- gether with some fundamental notions of rational mechanics, which were indispensable, and which will serve as an introduction to the regular study of this latter science. J. B. ^ "^■^ I CONTENTS. PAes Inteoduction 9 PART I.— DIFFERENTIAL CALCULUS. Section I. Mules of Differentiation, Algebraic functions of one yai-iable, 21 Transcendental functions of one variablei .... 27 Functions of two or more variables, 36 Implicit functions, 40 Section II. Successive Differeyitials* Maclaurin*s formula, 46 Taylor's formula, 49 De Moivre's formulas, 55 Maxima and minima, 56 Exercises on maxima and minima, 63 Values of functions which assume an indeterminate form, . 79 Section III. Investigatiom about Plane Curves, Tangents, normals, etc., . 85 Direction of curvature .90 Singular points, 93 Order of contact, osculation, 101 Measure of curvature, 104 Evolutes, .108 Envelopes, . . , . . ... . . .113 Elements of arcs, surfaces, and volumes, . . • .117 5 CONTENTS, Differentials with polar co-ordintftes, Spirals, PAOB 119 PART U.— INTEGRAL CALCULUS. Section I. Various Methods for Finding Iniegrcds Integration of elementary forms, , Reduction of differentials to an elementary form Integration by parts, Integration of rational fractions, Integration of binomial differentials. Integration by successive reduction, Integration of some trinomial differentials, Integration by series, Integration of trigonometric expressions, . Integration of logarithmic differentials. Integration of exponential differentials, Integration of total differentials of the first order. Integration of tlie equation Mdx -h Ndy = 0, Integration of other differential equations, Integration by elimination of differentials. Double integrals, Section II. Application of Integral Calculus to Geometry Rectification of curves, Quadrature of curves, . Surfaces of revolution, Solids of revolution, Other geometrical problems, Problems solved by double or triple integrals. Section III. Application of Integral Calculus to Mechanics Work, MoTement uniformly varied, Movement not uniformly varied, 132 138 142 144 151 158 161 168 165 169 170 172 175 182 192 197 199 204 208 210 213 223 . 231 . 287 . 238 CONTENTS. 7 PAGE Composition and decomposition of forces 241 Moments, 245 Virtual moments, 249 Attraction of a sphere on a material point, . . . 251 Centre of gravity, 256 Moment of inertia, 268 Curvilinear movement, 272 INFINITESIMAL CALCULUS. INTRODUCTION. 1. The Infinitesimal Calculus is exclusively con- cerned with continuous quantities ; for these alone admit of infinitesimal variations. A variable quantity is said to be continuous^ when it is of such a nature that it cannot pass from one value to another without passing through all the intermedi- ate values. All the parts of a continuous quantity are continuous : and, as all continuum is divisible, every part of a (Continuous quantity, how small so- ever it be, is still further divisible. In other terms, the division of continuum can have no end. 2. Infinitesimal quantities are sometimes con- ceived as resulting from an endless division of the finite. But this is not the real genesis of infini- tesimals ; for, in the order of nature, it is the infini- tesimal itself that gives origin to the finite. Thus, an infinitesimal instant of duration does not arise from any division of time; for it is the instant itself that by its flowing generates time. In like manner, the infinitesimaj length described by a moving point in one instant of time does not origin- ate in any division of length ; for it is the actual infinitesimal motion of the point itself that by its continuation generates a finite length in space. 9 10 INFINITESIMAL CALCULUS, Hence infinitesimals of time and of length are not mathematical fictions. They are true objective realities. Had they not a real existence in nature, neither the origin nor the variations of continuous movement would be conceivable. For the same reason we must admit that continu- ous action cannot produce acceleration except by communicating at every instant of time an infini- tesimal degree of velocity : and speaking general- ly, all continuous quantities develop hy inflnitesi- mat Tnoments. Hence the branch of Mathematics which investigates the relations between the con- tinuous developments of variable quantities, has re- ceived the name of Infinitesimal Calculus^ and its method of investigation has been called the infini- teHmal method. This method has been used by the best mathe- maticians up to recent times. Poisson, in the intro- duction to his classical Traite de Mecanique (n. 12), says: ^'In this work I shall exclusively use the in- finitesimal metliod. . . . We are necessarily led to the conception of infinitesimals when we consider the successive variations of a magnitude subject to the law of continuity. Thus time increases by de- grees less than any interval that can be assigned, however small it may be. The spaces measured by the various points of a moving body increase also by infinitesimals ; for no point can pass from one position to another without traversing all the iif^ termediate positions, and no distance, how small soever, can be assigned between two consecutive positions. Infinitesimals have, then, a real exist- ence: they are not a mere conception of Mathe- maticians." INFINITESIMAL CALCULUS, 11 3* Modem authors often define the infinitesimal as the limit of a decreasing quantity. This defini- tion we cannot approve. For the divisibility of continuum has no limit, and therefore cannot lead to a limit. This is so true, that even those authors confess that the limit — the absolute zero — can never be reached. On the other hand, infinitesimals, in the order of nature, do not arise from finite quanti- ties: it is, on the contrary, these quantities that arise from them. The origin of infinitesimals is dynamical ; for they essentially either consist in, or depend on, motion: and as motion has no other being than its actual becoming or developing, so also infinitesimals have but the fleeting existence of the instant in which they become actual. It is for this reason that Sir Isaac Newton conceived them 2iS fluxions and nascent quantities ; that is, quanti- ties not yet developed, but in the very act of de- veloping. This is, we believe, the true notion of the infinitesimal, the only one calculated to satisfy a philosophical mind.* So long as it remains true that a line cannot be drawn except by the motion of a point, so long will it remain true that an infini- tesimal line is the jfliuvion ot a point through two consecutive positions. An infinitesimal change may be defined, a change which is brought about in an instant of time. Now, the true instant is the link of two consecutive terms of duration : and it is obvious that between two consecutive terms of duration there is no room for any assignable length. Hence the fleeting in- stant has a duration less than any assignable — ' » ■ ' ' " ' ' * On the modern doctrine and method of limits see the note appended to No. 23. 1 2 INFINITESIMAL CALCUL US, length of duration, that is, it lias a dumtion strictly infinitesimal. And in the same manner, every other infinitesimal change is a link of two consecutive terms ^ or of two consecutive states ; for it takes place in an infinitesimal'instant. 4. But here the question arises : How can an in- finitesimal quantity be intercepted between two consecuiive points? -Consecutive points touch one another and leave no room between them. It would seem, then, that what we call " an infinitesi- mal" is not a quantity, but a mere nothing. We answer that a point in motion has always two con- secutive modes of being in space ; for it is always leaving its last position, and always reaching a fol- lowing position which cannot but be consecutive to the last abandoned.. Now, it is plain, that if the actual passage from the one to the other were not a real change, the whole movement would be without change ; for the whole movement is but a continu- ous passage through consecutive points. But movement without change is a contradiction. It is therefore necessary to concede that between two consecutive points there is room enough for an in- finitesimal change, and accordingly for an infinitesi- mal quantity. As a further explanation of this truth, let us con- ceive two material points moving uniformly, the one with a velocity 1, the other with a velocity 2. Their movement being essentially continuous, there is no single instant in the whole of its duration, in which they do not pass from one point to a con- secutive point, the one with its velocity 1, the other with its velocity 2. But the velocity 2 causes a change twice as great as that due to the velocity 1. INFINITESIMAL CALCULUS. 13 Therefore the ratio of the two movements is, at every instant^ as 2:1. But two absolute nothings cannot be in the ratio 2:1. Therefore the move- ments comprised between two consecutive points are not mere nothings, but are real quantities, though infinitely small. They are, in fact, fluxions^ or nascent quantities^ or, as the Schoolmen would say, quantities in fieri. Nor does it matter that these infinitesimals are sometimes represented by the symbol 0. For this symbol has two meanings in mathematics. When it expresses the result of subtraction, as when we have a--a = 0, it certainly means an absolute n/>thing: but when it expresses the result of di- vision, it is a real quotient^ and it always means a quantity less than any assignable quantity: but because it has no value in comparison with finite quantities, it is treated as a relative nothing y and is represented by 0. Thus, in the equation 00 the zero represents an infinitesimal quantity. This can be easily proved. For it is only continuous quantities that admit of being divided in infinitum : and, when so divided, they give rise to none but continuous quotients, because every part of con- tinuum is necessarily continuous. Now, the ab- solute zero cannot be considered continuous. Therefore the absolute zero c^p never be the quo- tient of an endless division. And in this sense, it is true, as the theory of limits affirms, that a de- creasing quantity may indefinitely tend to the limit zero, but can never reach it. On the other 14 INFINITESIMAL CALCULUS. ' - - .. ^ handy the above equation gives a = Xoo; and this does certainly not mean that the finite quantity a is equal to an infinity of absolute noth- ings. 6. We have said that infinitesimals have no value as compared with finite quantities. A few years agOj an American writer* was bold enough to maintain that this fundamental principle of infini- tesimal analysis is not correct. The principle, •however, has been admitted by the greatest mathe- maticians, and its correctness will not be doubted by any one who understands the real nature of in- finitesimals. The principle, says Poisson (loc. cit.), ''consists in this, that two finite quantities which do not differ from each other except by an infini- tesimal quantity, must be considered as equal ; for between them no inequality ^ how small soever, can be assigned^ ^ ; because the infinitesimal is less than any assignable quantity. Again, it is plain that the infinitesimal is to the finite as the finite is to the infinite. Now, the infi- nite is not modified, as to its value, by the addition of a finite quantity. Therefore the finite is not modi- fied by the addition of an infinitesimal. That the infinite is not modified by the addition of a finite quantity, can be assumed as an evident truth : but it can also be demonstrated. Thus, it is shown in Trigonometry that between the angles J., B, (7 of a * Mr. Albert Taylor Bledeoe in his Philosophy qf Mathematics^ where he (strives to prore that the infinitesimal method should be abandoned. Wo are afraid that philosophical readers will not consider his effort a saccess. INFINITESIMAL CALCULUS, 15 plane triangle there is the relation tan A + tan-B + tan(7= tan A tan jB tan (7/ and this equation, taking J. = 90^ jB=46^ (7 = 45°, gives which shows that the addition of a finite quantity does not modify the value of the infinite. We may draw from Arithmetic a still plainer proof of our principle. Dividing 1 by 3 we obtain i = 0.333333 . . . and multiplying this by 3, we obtain 1 = 0.999999 . . . In this last equation, if the second member be understood to continue without end, the difference between the two members will be an infinitesimal fraction — viz., unity divided by a divisor infinitely great. Now, we can prove, that, notwithstanding this inflnitesirrial difference^ the equation is rigor- ously true. For, let the second member of the equation be represented by x; then 0^ = 0.999999 . . . Multiply both members of this by 10 ; then lOa; = 9.999999 . . . = 9 + a;/ and from this, by reduction, we have This clearly shows that the equation 1=0.999999 . . . is rigorously true. It is plain, therefore, that an infinitesimal difference has no hearing on the value of a finite quantity^ and that no error is com- 16 INFINITESIMAL CALCULUS. mitted by suppressing an infinitesimal by the side of a finite quantity. 6. The notions above developed may suffice as a first introduction to the infinitesimal calculus. We have shown — 1st. That infinitesimals are not nothings, but ob- jective realities : 2d. That infinitesimals are not limits of decreas- ing quantities, but fluxions — that is, quantities in the act of developing, or more briefly, nascent quantities, whose value is less than any assignable value of the same nature : 3d. That infinitesimals may have different rela- tive values, and form different ratios : 4th. That an infinitesimal, whether added to, or taken from, a finite quantity, cannot modify its value. As to the different orders of infinitesimals, of which we slmll have to speak throughout our treat- ise, we have here simply to state the fact, that in- finitely great, and infinitely small quantities are capable of degrees, so that there may be infinites and infinitesimals of different orders, each infinite of a higher order being infinitely greater than the infinite of a lower order, and each infinitesimal of a higher order being infinitely less than the infini- tesimal of a lower order. How this can be, one may not find easy to explain, because both the infinite and the infinitesimal lie beyond the reach of human comprehension : nevertheless we know, not only from Algebra and Geometry, but also from rational philosophy, that such orders of infinites and of infinitesimals cannot be denied. We know that the species ranges infinitely above the indi- INFINITESIMAL CALCULUS. 17 vidaal, and the genns infinitely above the species. Substance extends infinitely less than Being, animal infinitely less than substance, man infinitely less than animal. From this it will be seen that the notion of an infinite infinitely greater than another infinite, is not a dream of our imagination, but a well-founded philosophical conception, familiar to every student of Logic, and admitted, implicitly at least, by every rational being. Let us, then, write the following series : X a? of If we assume 0^=00, it is plain that the first term will be infinitely greater than the second, the second infinitely greater than the third, and so on. The middle term 1 being finite, all the following terms are infinitesimal, and each is infinitely less than the one that precedes it. Hence infinites and infinitesimals are distributed into orders. Thus, if X be an infinite of the first order, a? will be of the second order, a? of the third, etc. ; and in like manner 1 will be an infinitesimal of the first order, X 1 1 -3 of the second order, -^ of the third, etc. 7. The problems whose solution depends on the infinitesimal calculus, are generally such that their conditions cannot be fully expressed in terms of finite quantities. Hence a method had to be found by which to* express such conditions in infinitesimal terms. The part of the Calculus which gives rules for properly determining such infinitesimals and their relations, is called the Differential Calculus. As, however, none of sucli 18 INFINITESIMAL CALCULUS, infinitesimals must remain in the final solutions, rules were also to be found for passing from the infinitesimal terms to the finite quantities, of which they are the elements ; and to effect this, a second part of Calculus was invented under the name of Integral Calculus. Of these two parts of the infinitesimal calculus we propose to give a substantial outline in the pres- ent treatise : and we shall add a sufficient number of exercises concerning the application of the Cal- culus to the solution of geometric and mechanical questions ; for it is by working on particular ex- amples that the student will be enabled to appreci- ate and utilize the manifold resources of this branch of Mathematics. PART I. DIFFERENTIAL CALCULUS. 8. Our object in this part of our treatise is to find, and to interpret, the relations which may exist between the infinitesimal changes of correlated quantities varying according to any given law of continuous development. Such a law is mathe- matically expressed by an equation between the variable quantities ; and it is, therefore, from some such equation that the relative values of the infini- tesimal changes must be derived. An infinitesimal change is usually called a differ- ential^ because it is the difference between two con- secutive values, or states, of a variable quantity. The process by which differentials are derived from given equations is called differentiationy and the equations themselves, by the same process, are said to be differentiated. Hence this part of infinitesi- mal analysis received the name of Differeniial Calculus. Differentials are expressed by prefixing the letter d before the quantities to be differentiated. Thus, dx = differential of a?, d{aaf) = differential of oaf. 9. When an equation contains only two vari- ables, arbitrary values can be assigned to one of them, and the equation will give the corresponding 19 20 INFINITESIMAL CALCULUS. values of the other. The one to which arbitrary values are assigned is called the independent vari- able, and the other, whose value depends on the yalue assigned to the first, is said to be a fwfiction of the same. Thus in the equation of the parabola, 2/* = 2pa?, if we take x as independent, y will be a function of x. When an equation contains more than two vari- ables, then all the variables but one can receive arbitrary values, and are, therefore, independent, whilst the remaining one will be a function of all the others. Functions are often designated as follows : y=f{^\ z = ^(^, y\ and _ 1 y—\/s=^s^, the diflEerential will be 1 1 I ^ 1 ^-=^ ^ ^ ds ds If 71=2, then INFINITESIMAL CALCULUS, 25 that is, the diffei^ential of the square root of a quantity is equal to the differential of the quantity divided by twice the radical. The preceding rules are sufficient for the diflPeren- tiation of any algebraic function of one variable. Examples. It is of the utmost importance that the student should at once familiarize himself with the above rules of differentiation, and test, by ex- amples, his practical knowledge of them. Let him work out the following : 1. y = ax^ -bx-}- ac, dy = (3a^— b) dx, 2. y = {a'+xy-b, dy = \:{a^-\'7?)xdx, 3. 2/ = «^ +- J dy= (^ccx -^)dXy a + x , 2adx 6. 2/ = 2 16. y =a ^ 1 + cot 5 « whence ^cot-*2/)=-j^.. sin s ds From 2/ = sec ^ we have found dy = hence cos" s ' sm s Vl — cos' 5 sec s -^sec' 5 - 1 ' whence ^2/ d(sec"' y) y Vy"- 1 T^ , - , , cos 5 ^«9 From y =cosec s we have found dy ^ ^-i — : ^ ^ sin s ' hence INFINITESIMAL CALCULUS. 35 ds^-dy^"^^'^ COS s sin* $ dv --dy ^ Vl — sin" s cosec s i^cosec* ^ — 1 ' whence dy tf(cosec~*y) = — yVy^-l From y = vers $ we have found eZy = sin s ds ; hence ^2/ dy dy sm s Vl — cos' s 4^1 — (1 — vers sf which reduces to dy tf(ver8~'y) = V2y - y' Finally, from y = covers s we have found dy = — cos s ds ; hence ds=- ^y ^ — cos s \fi - sin* s dy i^l — (1 — covers sY ' which reduces to d (covers" y) = — V2y^ EXAMPLES. 4 • 1 *^ J dx 36 INFINITESIMAL CALCULUS. re 4. 5 = COS"' , d5 = — adx a — X* {a—x) Vd'—^dx' 5. * = tan-'g), ^=-^^' 6. 5 = vers"'|-|j ^^ = — — /^ » ^ .;„ ..^Hil, , _ dx 7. 5 = sin"*— 7=—, ds = ,- ^^ . i/2 V^l -2x — x* ' s == sec —3 — - , ds= - 2x'-\' ""--4/T^r^ 9. s = tan-' i/|-=-22i^, cZ« = idyy y y 3. u — x* — 4r^, du = (Sa;"— Ay^'dx — 9>ocydy, 4. w = sin (a; + y\ du = cos (x + y) (dx + dy), 40 INFINITESIMAL CALCULUS. xdx + aydy + hzdz 6. M = iV-i-a^+5F', du = 1^0^ + ay' + 62' 7. w = 2tau-|, tftt=c?2tau-|+2 2^^^, -2 ^ z ^ zy Implicit Functions. 21. An implicit function is one whose value is only implicitly given in an unsolved equation. Thus y' — ^xy = a^ is an implicit function of x; whereas, if we solve the equation,, we shall have the explicit function y = x± Va* + ir*. When the function becomes explicit, its differen- tial is^ found by the rules already given ; but, as some equations cannot be readily solved, the func- tion may remain implicit, and its differential is then to be found by the following process. Let be the given function. Its differential will be f{x-\-dx, y + dy)-f{x, y) = 0, or, by adding and subtracting" the term/(ir+e&c, y)j /{x + dx, y)-f{x,2/)+f{x + dx, y-i-dy) -f(x + dxj y) = 0. INFINITESIMAL CALCULUS, 41 The first of these two differences expresses the differ- ential of the function with respect to x^ and the second exhibits its differential with respect to y. Denoting the first by ^ dx^ and the second by ^dy, we have Hence, the differential of an implicit function / (a?, 2^) = is obtained by differentiating it first with respect to x^ as if y were constant, then with respect to y, as if x were constant, and making the sum of the results = 0. Thus, from the equation a^y" + 6 V — a'6' = 0, we shall obtain ■^dx = 2Vxdx^ j-dy = 2a^ydy, and ZVxdx-^- 2a^ydy = 0. 22. By a reasoning analogous to the above it may be shown that the differential of an implicit function of three variables, as ' (!)• (g)' • ■ • be what y, ^ , -X 5 . • . become in this hypothe- dx dof sis. Then <^' = ^' it) = ^' (S) = ^'^' (g) = ^-^^^ • ■ • whence and, substituting these values in (1), ^=<^)+(i)^+(S)f+(S).4 +(S)m+--- (») This is Maclaurin's formula. In using it, it is necessary, of course, that the values attributed to x be such as will make the series convergent If a function is not susceptible of development by this formula, the formula itself will give notice of the fact ; for, in such a case, some of its constant factors will become infinite. EXAMPLES. 1. To develop y = sin or. We have dy d^y . . 2/ = sin rr, ^ = cosrr, ^ = - sm^r, d'y_ dx^ ' 48 INFINITESIMAL CALCULUS. 1)6I1C6 «=».(i)='.(S)=<>.(S)=-^.-- and substituting in the formula (5), emx = x- j^ + J 2 3.4.5 ~ 1.2.3.4.5.6:7 + * ' ' This series, being differentiated and then divided by dx, gives cos a; - 1 - ^ + 1.2.3.4 1.2.3.4.6.6 + ' • * 2. To develop y = (1 + ^)*- We have y = {l + xr,^ = n{l+xf-\ g = 7i(7i-l)(?i-2)(l + a;)"-», . . . II6IIC6 <^)=Mi)=»-(S)=»<'>-»^ and therefore (1 + xr = 3. To develop y = log (1 + ^)- We have 2/ = log(l + rr),g=-j^, d^p 1__ fg'y _ 1.2 INFINITESIMAL CALCULUS. 49 hence .«=»'(i)='.(g)=-.(S)=^.--- and therefore log(l+:r) = a;~2- + |--^+ . . . 4. To develop y = a*. We have 2^ = «^ S = ^'^ l<>g ^^ S= «" (log «)N da? ^ ^ da? ^ = a«aogay, whence w=''(i)='°«<^(S)=<'°«'">' and therefore a» = l+aoga)a. + ^^'^+^^+ . . . If we make a = ^, whence log a = log ^ = 1, then we have Taylofs Formula. 35. Let u^f{x) and t^' =/(;z;+A). Considering a? and h as two arbitrary parts of a certain line, it is obvious that, if the line receives an infinitesimal in- crement, the result will be the same, whether the increment be attached to the part x or to the part h. In other terms, the result will be the same whetlier the function i^'=/(^ + ^) be differentiated with 50 INFINITESIMAL CALCULUS. regard to a:, considering h as constant, or with re- gard to A, considering x as constant. In the first case, the differential coefficient of the function will, be -J— : in the second case it will be -^r- ; and there- dx dh^ fore we shall have du' _dml_ ,.. d^^dh'''' ^^ This equation will afford us the means of de- veloping the function u' =f{x-\'h) into a series ar- ranged according to the ascending powers of h^ with coefficients that are functions of x alone. Let us assume a development of the form u'=zPJ[^Qh-{-RJi' + Sh'-\-TJt'+ ... (2) in which P^ Q, H^ . . . are functions of x alone. Differentiating (2) with regard to x, and dividing by dXy we shall find dx dx ^ dx ^ dx ^ dx ^ dx ' ^ ^ then, differentiating (2) with regard to 7^, and divid- ing by dhy -^' = g + 2i2A + 3Sh' +4Th' + . . . (4) Now, by (1), the first members of (3) and (4) are equal ; hence their second members are also equal, and the coefficients of like powers of h in those second members are equal. Therefore INFINITESIMAL CALCULUS. 51 But P is the value u of the function when A = ; and therefore dP = du; hence we shall have 1 eZ^it These values substituted in the equation (2) give '^-'^^d^^^d^''/ 2 +^-'2:3 d^u A* ,. + ^•2:3:4+ • • • ^^^ This is Taylor's formula. The values attributed to X and Th must be such as will- render the series crni- vergent EXAMPLES. 1. Let u=^af^\ then i^' = (re -|- A)* ; and we have ^ = n (TO - 1) (» - 2) a!»-», . . . hence u' = {x + Jif = of" + naf" h -{-''^^^^—^ af"-^ h* 2. Let z^ = log x; then z^' = log (a? + A); and we have du 1 d^u 1 ^?^ 2 ^*f^ 2.3 eZa? ic ' rfa?* x"" ' 6?rc" rr' ' dx*^ X' whence • • ^ 52 INFINITESIMAL CALCULUS. 3. Let u = (f ; then w' = «*** / and we have ^_^ ^-^ ^-^ dx~^' daf~^' da?~^' ' ' ' whence In this equation make x = and h= 1. Then we have _ 111 or e = 2.718281828469 . . . , which is the basis of the hyperbolic logarithms. 4. Let u = BiaXj and u' = sin (a? + y) ; then we have du d^u . d*u -=- = cos Xj '3-5 = — sin X. -5-5 = — cos rr, , . . dx daf ' dx^ ' whence y % u' = sin (rr + y) = sin rr + y cos rr — ^ sin a? If we change y into — y, we shall have also sin (^ — y) = sin re — 2/ cos re — ^ sin x X y" ' I V • • • t • INFINITESIMAL CALCULUS. 53 Scholium. Taylor's formula may be used for the development of a function u =/(a;, y) of two independent variables. If we begin by giving to a; an increment h, we shall have /(«; + *, y) = « + ^A + ^^ + ^^+ . . . and in this equation, when we give to y an increment k, the first member will become /(a; -k-h, y -k- k\ and in the second member u will become du. dSkk* dht k^ dy ^dy^ %^dy^2.% j-h will become dx u\ and u>u" ; hence the two series miisfc be both negative. If t^ is a minimum, then u-=(S)j-(su;+---. ^-1 j . If this INFINITESIMAL CALCULUS, 61 term be negative, Ua will be a maximum : if it be positive, Ua will be a minimum. -3—, 1=0, then the two seiie? would reduce to '_ - (^\ K _L /^\ *' ^ ^~ V<^a^/, 2.3 "^ UW. 2.3.4 " ' * * ' ^ -^= + (^'l 2:3+ fel 2:3^+ • • • ' and thus tliey would again have different signs. Hence there could be no maximum and no minimum, unless (^-^i) =0. In this case the two series would begin by the term \^i) 5-0-7 , which, if positive, would give a minimum, and if negative, a maximum. If this term also were to become = 0, we would have to proceed as before with regard to the subse- quent terms of the series. From all this we may draw the following conclusion : If the first differential coefficient which does not become =0 is of an uneven order^ the two series have opposite signs^ and there is no maximum or minimum. If the first differential coefficient which does not become = is of an even order, the two series have equal signs, and there wiU he a maximum when their sign is negative^ and a mini- mv/ffi when their sign is positive. 39. The investigation of maxima and minima may often be simplified. Thus a constant factor 62 INFINITESIMAL CALCULUS, tliat affects the whole function can he suppressed du in the diflferentiation ; for, since we have ^— = 0, the result is independent of any such factor. So also, if we have a function of the form y = Va^x - bafj we can make y^=u = a^x— 6a?', whence — = a* — 2hx^ and -3-5 = — 26. Making ^ = 0, a* we have rp = prr • This value makes u a maximum : 2b hence it makes y^ a maximum, and consequently also y a maximum. By this artifice we can differ- entiate the function without taking notice of the radical sign. And, again, we may simplify oi)erations hy tak- ing the logarithms of the quantities to he differen- tiated. Thus, if we have a function _ {x-\) (^-2) 2^-(a? + l)(^ + 2)' passing to logarithms, and making log y='U^ we shall have t^=log (a:~l)+log (a?-2)-log (;r+l)-log (^+2), and du _ 1 . 1 1 l__o • dx'^ x'^^'^ x-2 x + l"^ x + 2~ ' from which we obtain a: = ± V~2. It is obvious that, when this value of remakes u a maximum or a minimum, y also will be a maximum or a minimum. When the first differential coefficient is a pro- duct of two or more factors, and one of these fac- tors becomes = for a value of x corresponding to INFINITESIMAL CALCULUS. 63 a maximum or to a minimum, the second differ eTir Hal coefficient can be obtained witJiovi differen- tiating the other factor s^ as in the following ex- ample. Let dU T^ ^ T> P, Cj and R being functions of x. The regular dif- ferentiation would give * Now, if the factor jB, for instance, becomes = for a value x = a^ \t is evident that the differential will reduce simply to \da^')a~'\^^dx)a ' Hence it will suffice, in such a case, to multiply the other factors by the differential coefficient of the factor which becomes = 0. Exercises on Maxima and Minima. 30. The application of the preceding principles to the solution of problems is not difficult, though the student may, at times, experience some difficulty in finding out the mode of expressing the particular function which is to be worked upon. A few ex- amples will sliow how the difficulty may be prac- tically overcome. I. Required the dimensions of the maximum cylinder that can be inscribed in a given right cone. 64 INFINITESIMAL CALCULUS. Let A VB (Pig. 1) be the cone, and suppose a cylinder i9scribed. Let VG = h, AC=zr, VO = y,DO = X. The volume V of the cylinder will be expressed by V=7:x'{h'-y). But from the similar tri- angles A VC and D VO we have x:y::AC: VC::r : hj and hx ^ r Substituting this value of y in the preceding ex- pression, we have V= — re* (r — x). Hence g='^(2..-an^^=?(ar-te). dV Placing -J— = 0, we find the roots a? = and x = 9r — . The first value placed in the expression of drv da? makes it positive, the second makes it nega- tive. Hence .r = corresponds to a minimum, and 2/* a?=-7r^ to the maximum required, which will be F= . Its altitude is =^ , whilst the radius of its base is = — . IISTINITESIMAL CALCULUS. 65 FlQ.t II. In the line CO' (Fig. 2) which joins the cen- tres of two spheres y to find th^ point from which the greatest portion of spherical surface is visible. Let A be the point sought for. Draw the tangents AJIf and AM\ the radii CM = r, CM' =r\ and c'^ the lines MP, MP' perpendicular to CC Make CC = a, and AC=x. The portion of surface visible from A on the right hand is one-half of the sphere, minus a zone of the altitude P(7, or 27cr' - 2irr X PC ; and the portion visible on the left hand is one-half of the surface of the other sphere, minus a zone of the altitude P'C\ or 27rr" - 2Tvr' X P'C Calling s the total visible surface, we have s^2n ij^'^r'^ ^rxPC-r'X P'O). But PC \ rwr : x, and P'C : r'::r' : a —x; lipiice PC = ^- , andP'^'= — — ;and therefore X ^ a — x' s = 27r(r' + r" - ^' ~ -^) ; \ ' X a — xf ' whence 66 INFINITESIMAL CALCULUS, ds dx ds dx x ~ W (a - xY) ' {a - xy and making ;^ = 0, we shall find x=ia Vr' Such is, then, the distance from the centre C to the point A that satisfies the condition expressed in the enunciation of the problem. III. Through a point P (Fig. 3) a straight line is drawn meeting the axes OX and OT at A and B respec- tively. Find the least length ^ that this line can have. Let OM^=.a^ and PM=b be the co-ordinates of the given point P, and make the angle PAO = ^. Then h PA = , and PJS = a sin 1? ' ^"^ " cos t? hence, making AB = u, we shall have ^=^^zr^ + a whence du d^ sin I? ' cos ^ ' b cos & g sin ^ sm^ + cos'* ' du Making ^tk = we shall find d» tan ^ =(iy- INFINITESIMAL CALCULUS. 67 From this last equation we obtain cos and therefore s (7 r sin 17 V This is the minimum required. IV. To find the least cone that can be circum' scribed oJbou ^ a given sphere. Let S0=7i (Fig. 4) be the altitude of the cone, AC=Ii the radius of its pig^^ base. Then the expression of its volume V will be Draw the radius OP = r to the point P where the element B8 of the cone touches the . sphere. The similar triangles 8CB and SOP give us CjB:B8::0P: OS, that is, H: VH' + h'y.rih-r; whence, by developing and reducing, we find D«_ r'h This value of JJ' substituted in the above equation, gives V= 3 ' h-2r' whence 68 INFINITESIMAL CALCULUS. dV dh _w' {h -2r)2h- h _ (A - 2ry 0, from which we obtain h = 4r, and therefore -B" = 2r'. The volume of the minimum cone will therefore 8;rr* be F= As the total surface of this cone is S=z87tr*j we see that as its volume is twice that of the sphere, so also is its total surface twice that of the sphere. V. A man being in a boat a miles distant from the nearest point of the beach^ wishes to reach in the sh/yrtest time a place b miles from that point along the shore. Supposing that he can walk m miles an hour^ Met pull only at the rate of n miles an howTy required thepfax^e where he must land. Let AB = a{¥ig. 5), BC=b, BD = x, D being the point where the man must land. The time employed in rowing from A to i> will be \ Va^ + af , and the time em- ployed in walking from D to B will be ^ (& — x) ; so that the total time T employed in the journey will be ^__ v'a'H-^ jb — x n m Differentiating, we find for the minimum dT_ X dx^ n Va' + af ^ =0, anda: = an j^m' - 71' INFINITESIMAL CALCULUS. 69 This value of x gives the distance BDy and the point D wl^ere the man must land. The shortest time will be mn Make a=4, & = 8, m = 6, n = 3; then a? = 3, and r=2*40^ VI. A triangle has a base b and a perimeter 2p. Whai must its second and third side be, that the triangle be a raaadmum f Ijet the second side be denoted by x ; the third will then be 3p — 6 — a?. Its area will then be ex- pressed by the equation J. = >/p (^ — 5) ( jp — rr) (ft 4- re — jp). It will be found by the ordinary process that A is a maximum when x = -^-^ — / the triangle being isosceles. VII. To find the dimensions of the maximum solid cylinder whose total surface is S. Let X be the radius of the base, and y the alti- tude of the cylinder. Its total surface will then be expressed by and its volume by F=;ra?'.y, or, eliminating y, by V=nc^^-~^ = k^8x-2r:a*)', whence dV dx = \{8- 6ffar') = 0, x = >|/g , y = 2^^. 70 INFINITESIMAL CALCULUS. or yr=2x. Hence Fis a maximum when its alti- tude is equal to the diameter of its base. VIII. To find the greatest isosceles triangle that can he inscribed in a given circle. It will be found that it is an equilateral triangle. IX. To fl,nd the smallest isosceles triangle that can be circumscribed about a given circle. It will be found that it is an equilateral triangle. X. To find the greatest rectangle that can be in- scribed in the ellipse whose semi-axes are a and b. Its base will be a V% and its altitude b V2. XI. To find the greatest cone that can be cut from a sphere. If r is the radius of the sphere, the altitude of the cone will be -^ , and the radius of its base — \/2. o o XII. To find the greatest segment of a parabola that can be cut from a right cone. If h is the altitude of the cone, and r the radius of its base, then the axis of the segment will be I l/F+F. XIII. To find the maximum parabola that can be inscribed in an isosceles triangle having the altitude h and the base b. Let VH— X, and PH = y (Fig. 6) be the co- ordinates of the point P, where the parabola touches the side CA of the given triangle. The area of the parabola will be ^ = j VD x DK, INFINITESIMAL CALCULUS, 71 As Cy = YH=x, we have YD = h-'X. On the other hand, DK' : PW\\ YD : YH, or i)ir'' -.y'wh-x'.x, and therefore =v- i)ir "--^ a; To eliminate y, we have from tlie similar triangles ADQ and PHG PH: AD:. OH .CD, OT y : ;^:: 2a; . h, or y = -^; hence Accordingly, the equation of the problem becomes A 46 r-7T A=^Vx{h--x)\ Hence dA_2h^ {h-xY-Sxih-xY dx~37t' |/5"(A - xY ~ ' and a? = - . Consequently YD = ^ ^ and i)^= XIV. To find the minimum parabola that can be circumscribed about a circle. Let r be the radius of the circle (Fig. 7), YII= x, Pff=2y the co-ordinates of the point P of contact, TM. = a, and AB = b the terminal co-ordinates, 72 INFINITESIMAL CALCULUS, and 2p the parameter of the parabola. The area will be J. = |a6. Now, a=VH+HO+ Fig.7 _^B OA = X +^ + ^? ^^^ ^' = 2^a. On the other hand, the triangle POff gives y^ z=r* -^p* = 2pXy whence x = - / and therefore ^ a= 2p 2p -fr+^ = r'+ 2pr + jP* _ {p + ^)' . 2p ~ 2p ' also 6' = (p + r)*, or 6 =^ + ^' Substituting these values of a and 6 in the expres- sion of J., we have whence ^i? "" 3 * i?* 97* and consequently 2p = r. Hence am—, and 6 = 3r "2 • XV. ^ cone has a total surface S. What must its dimensions he that its volume he a maximum f Let h be the altitude of the cone, and r the ra- dius of its base. Its total surface will be 8=7:r* 0, + Ttr Vh^ + /•', whence h=-Vsrz:2S^r\ INFINITESIMAL CALCULUS. 73 On the other hand, the volume will be Hence F=- Trr* X ^ = g V'/S' - 2/8!^r'. dV_l S' -- 287rr' - 28717^ _ dr~3' i/S' - 2/8irr* ~ ' which gives r=^^A/-^ and hence h=: y — . XVI. To find the maximum cylinder that can he cut from an oblate ellipsoid of revolutions whose semi-axes are a and b. The radius of the base = ai/^ , and the altitude XVII. What value of x will make u = r— ; — -^-—^ l-^X — XT a minimum? The function is a minimum when a? = J. XVIII. Through the focus of an ellipse two chords are drawn at right angles. Find when their sum will he a maximum^ and when a mini- mum. The solution will be reached through the polar equation of the curve. XIX. Show that u = sin X (1 -f cos x) is a maxi- mum when X = - . XX. To find the least ellipse that can he de- scribed about a given rectangle. 74 INFINITESIMAL CALCULUS. Let 2a and 26 (Pig. 8) be the sides of the given rectangle, and let AY + B^x' = A^ff be the equation of the ellipse. In all ellipses we have y' : y'" :: A' - a^ : A* -a;"; hence, if we take y=-B and 2/' = ft, we shall have a; = 0. nd = a, and B':b'::A*:A*-a\ or 5 = Ab VA' - a' The area of the ellipse is ^ = ttAB. Substituting for B its value, A' E-7th VA' - a' ' hence d:E_ j,2A{A'-a')-A* dA = 7lb = 0, V (A* - aj which gives A = a V2. And this value substituted in the expression of B gives B=b V2. Accordingly the area of the ellipse will be E = r.a\^XhV2z= 2nah. ^ ScHOLiuu. The theory of maxima and minima, as above developed (No. 28), can be extended to the Investigation of the maxima and minima of any function of two independent variables. Let the function be at its maximum or minimum, and let u' =f(x — K y — k\ and u" =f(x + ^, y + k\ INFINITESIMAL CALCULUS, 75 be the values of u immediately before and immediately after the maximum or minimum. Then by Taylor's theorem, as extended to two variables (No. 26, Seholium\ we shall have dx dy dx* 2 docdy dy* 2 „ du , du, d^u h^ d^u ,, d^u k* dx dy dx^ 2 dxdy dy^ 2 which may be written as follows : ti' — 1* = ^KTx''^dyn^-A^^^^''d^y''^^dii^n'''^' u" - -M = rfd; dy ) % \dx^ dxdy dy^ ) Now, if w be a maximum, then u > t/', and u > w", and the two series must be negative. If w be a minimum, then u )' which, if negative, will indicate a maximum, and, if positive, a minimum. Making dx^ •* ^' dS5y " ^' ^ ■*■ ^' the trinomial will take the form Ah? + '^Bhk 4- C3fc«, or -^(^41*5). or, by adding and subtracting -j, and factoring, Now, since A and k are arbitrary, and independent of A and J9, we cannot assume r; = — — ; and therefore the first term ( r + -j ) of the factor within the brackets cannot be = 0, and is always posi- tive. As to the second term, — -r^ — , it is easy to prove that it cannot be negative. For as the sign of (2) must remain unchanged for all the small values of the arbitrary constants K and ky it follows that the value of ttie expression (2) must not pass through zero. But, if we had AC^ ^ < 0, we might choose for h and k such arbitrary values as would give (li BY Kk^A) = A^ that is, (2) would pass through zero. Hence the assumption AC — B^<0 is inadmissible ; and we must therefore have either AC — B^ > 0, or AC — B^ = 0; and thus in both cases the factor within the brackets will be positive. Hence the sign of (2) will be the same as that of the other factor A, It follows that the existence of a maximum or a minimum cannot be inferred from (2) unless either AC> B^, or AG = B^ ; that is, unless INFINITESIMAL CALCULUS. 77 dx^ ' dy^ ^ \dxdy) * ^^ dx^ * dy' ~ \dxdy) ' ^^' The values of x and y which ought to satisfy eitlier the oue or the other of these two conditions must, of course, be taken from the equations (1). d^u The conditions (3) show that the differential coefficients ;^ and dx j-^ must always be either both negative or both positive ; and as the sign of (2), or of the trinomial, is always the same as that of dH dH T-j , it is plain that when -p, is negative the function u will be a maximum, and when ^-^ is positive the function u will be a mini- mum. But let the student remember that, although the existence of a maximum or minimum cannot be tested by (2) when the trinomial is = 0, yet even in this case there may be a maximum or minimum : but it must then be determined by the sign of the fourth differential coefficients, after having ascertained that the third differential coeffi- cients, which have opposite signs, reduce to zero, as the theory (No. 28) requires. Example T. A cistern which is to contain a certain quantity of water is to be constructed in the form of a rectangular parallelopipe- don. Determine its form, so that the smallest possible expense shall be incurred in lining the internal surface. Solution. Let a^ = its content, x = length, y = breadth, and therefore — = depth. The total surface u will be xy /v3 /«3 u=:xy ~{-2 ~ +2— =:a minimum. Differentiating first with regard to x, then with regard to y, we find du 2a^ _ du 2a'* ^ dx x^ dy y^ hence x^y = 2a^ = y^x, and x = y = a\/2 ; and therefore the base must be a square. The depth will be a» a3 \/^ 1 B/ii 78 INFINITESIMAL CALCULUS. and therefore the depth must l)e equal to half the length or breadth. Since the first of conditions (3) is satisfied, and u is a minimum. Example II. Find the values of x and y which shall make the function u = x^ + y* — iaxy^ # a maximum or a minimum. Solution. Here we find — = 4a;' — 4ay' = 0, — = 4y* — Saxy = ; hence aj3 = ay\ y» = 2ax, x^ = 2a% x^ = 2a* x= ± a>v/2, y* = 2a«y'3 = a»y8, y= ± a^S. And again, ^-^ = 12«' = 24a« — = 12y^ - 8ax = 16a« V2, and therefore the first condition (3) is satisfied ; and as the sign of -^ and -T^ is positive, x= ±a\/% and y— ± a^8 make u a dx^ dy^ ^ minimum. du ^ . du In this example, the equations — = and — - = are also satis- fied by taking a: = and y = 0. With these values of x and y we find ^TT = and 5—^ = 0. And since in this case the third differ- dx^ dy^ ential coefficients are and the fourth differential coefficients d*u _. d*u ^ = 24. ^-, = 24. •« INFINITESIMAL CALCULUS, ^9 are positive, we conclude that the values a; = and ^ = correspond to another minimum. Remark, When u is a function of three independent variables, the conditions of its maxima and minima are determined by a process analogous to the preceding, but which is based on the extension of Taylor's formula to a function of three variables, and is too long to be inserted here. The result, however, of such an investigation is simple enough. If a function has any maximum or minimum, it must give ^ — ——0 ~— dx'' * dy~ * dz ~ * and the values of x, y, z found from these equations must satisfy the condition I dx^ • dy* \dxdy) \ (dx^^ dz^ \dxdz) \ / d^u d^u d^u d^uy \dydz dx^ dxdy ' dxdzj ' The function will be a maximum if tlie two factors within brackets in the first member of this inequality are negative ; but a minimum if they are positive. Values of Functions which assume an Indetermi- nate Form. 31. It sometimes happens that in giving to the variable a certain value, the function assumes one of the forms 00 ^ 1 0' «^ ' ^ ' ' OXoo Thus the fraction -^ — - , when x=l. takes the form - , though its real value is 2. The real values of 80 INFINITESIMAL CALCULUS. functions that assume the fonn - can be found by the following process. Let z and y be functions of a?, and let w = - be- come - when a; = a. Clearing of fractions, and differ- entiating, we have udy'\-ydu = dz; but when x = a, the term ydu disappears. Hence {udy)a = {dz)a , or Accordingly, the value of u^ when it takes the form ^toTX=±a, will be found by differentiating sepa- rately the numerator and the denominator of its ex- pression, and substituting in the resulting fraction the value x = a. n (D^were again of the form ^ we would ap- ply again the same process of differentiation, and we would obtain ^— Wa' and, if necessary, we might continue the same pro- cess until a determinate value is reached. Thus X ~~ sm X the fraction , — becomes ^ when a? = ; but by X \) the process just explained we successively obtain INFINITESIMAL CALCULUS. 81 V x — sin a;" | __r i — cosrc "] _ rsin a? "] __r cos x l _1 When the function, for a certain value of a?,^s- sumes the form ^, or the form x oo , or the form 1 0X00 , its real value may be found by first reduc- ing it to the form , and then applying the process above explained. The reduction is easily obtained by remembering that oo = :; • Sometimes this re- lo&r X duction is not needed. Thus the function — ^ — , x—a which takes the form ^ when a; = oo , will give im- mediately riog^-1 ^rn Lx - aj« Ll.. = 0. The form oo — oo may also be reduced to the form ^. For let V and w be two functions of x^ which for a certain value of x become infinite. Then the function u:=v — w becomes oo — oo f or that value of X. But we have and when ij = oo and t^ = oo , the function will take O in the form ^ , provided we have 1 = 0. If this 82 INFINITESIMAL CALCULUS. condition were not fulfilled, then the equation u = t) (l j would make the function infinite. Assume u=^^ sec X — X tan x; when x = ^j the function takes the form oo — oo . But we may write (^)r6 sec x — xtdLUx and from this we shall obtain {u\ = 1. 33. The indeterminate forms 0^, oo°, 1*, can be reduced by the following method. Let v and w be two functions of x of such a nature that, when x=ay they cause the expression u = v^ to assume one of the forms 0°, oo °, 1*. Since d = ^®»'' {e being the base of the Napierian logarithms), we shall have ij* = ^*^^®»^ Now, the exponent w log v in each of the three proposed cases takes the form X 00 , which can be reduced to ^r , as we have ex- plained. EXAMPLES. ^- Llog (1 + ^Mo" ^' INFINITESIMAL CALCULUS, 83 3. [x log x\ = 0, r x" tan X "I _ ;r' Ll + t^^ ^Ji ~ 4' 5. [(l-^)tan^]=-, a. [taoS.og(.-?)X=?, » can.-. 10. r_i_i =, \jx cot a? Jo 84 INFINITESIMAL CALCULUS. SECTION III. INVESTIGATIONS ABOUT PLANE CURVES. 33. Let y=if{x) be the equation of a plane curve MPL (Pig! 9), and let rr = 0^ and y = PA be the co-ordinates of a point P of the curve. Draw PT' tangent to the curve at P, and PX' parallel to the axis OX ; and make the angle T'PX' = ??. Let the point Q be consecutive to the point P. The infini- tesimal increment PQ of the curve entails an increment AB of the ab- scissa x^ and an increment QR of the ordinate y. If then we represent by s the portion MP of the curve, then PQ=ds, whilst AB^dx^ and QR = dy. Now, we have rr AB = PQ cos i>, QR = PQ sin i?, PQ" — PR^-\- QR", hence dx=:ds cos ^, (?2/ = ds sin tf, e?5 = Vdaf + ^2/% and "=- =cos ??, 3^ = sin ^, 3^ = tan d. a* a^ c«c INFiyiTESIMAL CALCULUS, 85 As the tangent P J' is but a secant which meets the curve at two consecutive points, it follows that the tangent and the curve have a common infini- tesimal element PQ^ and that the angle which the element PQ of the curve makes with the axis OX is identical with the angle ^ made by the tangent at P with the same axis. The trigonometric tangent of the angle i? is taken as a measure of the slope of the curve at the point P, and, as tan ?? = -3^ , the differential coefficient of the ordinate of any point of the curve is the measure of the slope of the curve at that point. Tangents^ Normals^ etc. 34. The equation of a straight line passing through two given points of a curve, whose co-ordi- nates are x\ y\ and x% y% is When the two points are consecutive, as P and Q (Fig. 9), then y" — 2/' = ^y\ ^^^ x" — x' = dx'; and the equation becomes This is the equation of the tangent to the curve at the point P, whose co-ordinates are denoted by Making y = 0, we find for the point jT, where the tangent meets the axis OX, 1 86 INFINITESIMAL CALCULUS. X=iX — y The Bubtangent AT i^ evidently = re' — a; = y ,dx^ 36. The normal being perpendicular to the tan- gent at the point of contact, its equation can be de- rived from that of the tangent by substituting -0 instead of + ^ Hence 2/-2/=-^,(^-^) is the equation of the normal to the curve at the point {x\ y\ Making 2/=0, we find for the point N^ where the normal meets the axis OX, The subnormal AN is evidently equal to re — a?' -^y dx'' 36. An asymptote to a curve is a line that con- tinually approaches the curve and becomes tangent to it at an infinite dis- y/ Fig. 10 tance. Such a line will, of course, cut either one or both the co-ordinate axes at a finite . distance from the origin. Let the straight line A T (Pig. 10) be an asymp- tote to the curve LP. Since A T^ when infinitely A o L JT" prolonged, is a tangent to the curve, its equation will be of the form INFINITESIMAL CALCULUS. 87 , , 1 _M«_— ■_!■ ^M ■■ I I — TT m— i—T r-*l \ ^m§ x' and y' being the co-ordinates of the point of con- tact infinitely distant from the origin. This equa- tion, when 0? = 0, gives and, when 2/ = 0, it gives x=:x' — ^-7 y' = OA. If the values of OA and OB obtained from these equations are both finite, the asymptote will inter- sect both co-ordinate axes ; if one of the two values is infinite, the asymptote will be parallel to one of the axes ; if both values are zero, the asymptote will pass through the origin, and its direction will be determined by the value of ~ . If both OA and OB are infinite, the curve has no asymptote, as no place can be found for it in the plane of the co- ordinate axes. 3*7. Let us inquire, for example, whether the curve g/* = 2a? + Saf has any asymptote. We find by differentiation dy _ l+3x _ l + Sx ^~ y '^±V2x + Sx'' This value being put in the expressions for OA and OB will give, after reduction. 88 INFINITESIMAL CALCULUS. 0A=^^^-^, 0B= ^ l + ar' ±V2x + '6af' And now make a? = oo . Then 0A=^\, 0B=±— . 3 ' ^4/3 Hence the curve has two asymptotes, one of which 1 1 intersects the axes at the distance x= ^^, y= —711 1 1 and the other at the distance a;= — -, v = ^ • 3 V3 Again, let us inquire whether the curve y = logx has any asymptote. Here we have dx X ' hence OA = x {1- log x\ OB = log :r - 1. The assumption y = co gives both OB and OA infi- nite, and cannot correspond to an asymptote ; but the assumption y = — co gives a? = 0, whence we obtain (No. 31) OA = [x{l - log x)]o = ""^ "" *^^ a; = Mo = 0; a?' and therefore the curve has an asymptote which passes through the origin, and makes with the axis OX an angle whose tangent is ;^ = - = 00 ; that is, OaC X \ INFINITESIMAL CALCULUS. 89 ■ ■■ !■ I IIIM — - - — ■ ' ' ' ■ ■ — — * the asymptote is at right angles to OX, and co- incides with the axis T. Let us inquire also whether the curve y = tan x has any asymptote. Here we have ^=: -1- = 1 + tan'ir = 1 +2/'/ dx cos' re ' ' ^ hence 0^ = tan-i2/-j^, 05=tanaj-^- Make y = vi. Then 0-4 = 5,05=00. Hence the curve has an asymptote parallel to the axis or, at a distance - from the origin. As the ■ TT value 2/ = 00 corresponds not only to ^ = 05 hi^t also to a? = g- , = 2" 5 = 2^ ' follows that there will be an endless seriei? of asymptotes, all parallel to the axis OF, and all at a distance t: from each other. To find whether the parabola y^ = 2px has asymptotes, let us put its differential coefficient -^ = - in the expressions for OA and OB. We dx y shall find, after reduction, OA:rzz- X, 0B=^' Making y = 00 , we shall have also a: = 00 ; and thus 90 INFINITESIMAL CALCULUS. both OA and OB will be infinite. Hence this curve has no asymptotes. In tlie hyperbola represented by the equation ay - 6 V = - a^V we have dy __ Vx dx ~" a^y ' hence, by substitution and reduction, 0A = ^* X* 0B=-'^ y Making y = ± oo , we shall have also ar = ±00; and therefore OA = and OB = 0. Hence the hy- perbola has two asymptotes passing through the origin of co ordinates, and making with the axis OX two supplementary angles, of which the acute one corresponds to x and y affected by equal signs, and the obtuse one to x and y affected by opposite signs. Direction of Curvature. 38, Let Jl/, iVi P (Fig. 11) represent three con- secutive points of a plane curve y =f{x). If we draw the chord MP, and if^ the point iV lies above the chord, the curve is said to turn its concavity downwards ; if the point JN' were to fall beneath the chord, the curve would be said to turn its con- vexity downwards. Let ?/, y\ y" be the equidistant ordinates of the points M, N^ P. Since y' is consecutive to y, and y'^ consecutive to y\ wa have Y Fig. 11 1 3 J f ^=^ M / ^ .—- _ — / i E 5 C : X J INFINITESIMAL CALCULUS, 91 y' — y-^-dy, y" = y' + dy' = y + dy+dy\ If the curve is concave downwards, as it is in our diagram, then BN> BQ. But BN=y\ and BQ = i (S/ + yO ; therefore 2/' > kiy + l/\ or 22/' >y + y% that is, 2{y ^dy)> y + y -{-dy + dy' and, by reduction, ^y > ^2/'> or e?2/' — dyOy d^v or e?'2/ > 0, and ^ > 0. Accordingly, whenever the second differential co- efficient of the function is n^gative^ the curve is concave downwards ; and whenever the same co- efficient is positive^ the curve is convex downwards. If, in the equation of the curve, we take y as in- ^ dependent, we can show, by a reasoning similar to the above, that the curve turns its convexity or its concavity to the left^ according as the second dif d'x fereniial coefficient ^— is positive or negative. 39. If, for a certain value of a?, the first differen- tial coeflBicient -^ becomes = 0, and if the second d^9j differential coefficient ^-^ be negative, that value of d^v x will answer to a maximum^ whereas if ~^ be ^ da? 92 INFINITESIMAL CALCULUS, positive, that value of x will answer to a minimam, as we have seen (No. 28). It is plain, then, that any maxima and minima can be graphically ex- hibited as ordinates to the culminating points of a curve. . When ^ = 0, the tangent to the curve at the point of the maximum or of the minimum is par- allel to the axis of x. As, however, there may also be a maximum or a minimum for ^ = x , tliat is, dx for -5- = 0, the tangent may also be parallel to the axis of y. Thus in the ellipse (Pig. 12) whose equation is y' = --,{2ax~x'\ we have dy dx = ± b(a — x) a V2a^ — x^ n^ = 0, then a: a. and y=:±b, the value + 6 being a maximum, and the value —6a mini-^ mum. Tm dy lA^ dx ^ ^ ^ ^=^> t^®i^^=0, and 2air-a?' = 0; and acc6tf dingly x = %oxx = 2a, the first value being a minimum and the second a maximum. The double sign which affects the expression for ^, equally aflfects the expression for ^; and INFINITESIMAL CALCULUS. 93 therefore while oue half of the curve turns its con- cavity downwards, the other half turns it upwards. The value -^ = expresses the fact that the tan- gents at the points B and B' are parallel to the axis OX, and the value -^ = (» expresses the fact that the tangents at and at A are perpendicular to OX, that is, parallel to T. Singular Points. 40, A singular point is a point at which the curve presents some peculiarity not common to other points. Four classes of singular points are 'particularly remarkable \ points of inflection^ cusps ^ multiple points^ and conjugate points. A point of inflection is a point at which the cur- vature changes from concave downwards to convex downwards, or vice-versa. This change requires that the second differential coeflBicient of the function should change from — to + > or from + to — ; hence, at a point of inflection, the second differential co- efficient must be either = or = oo . If, then, the co-ordinates of the curve are such that, for values immediately preceding and following, the second differential coefficient has contrary signs, they are the co-ordinates of a point of inflection. Thus, the curve gives g = 3(a.-a)',g = 6(:r-a). When x = a, then ~ = 0. When x a, then -t— > 0. Henoe at dx x=a there is a point of in- flection, whose co-ordinates are rr = a, y=^h ; and since, dx' at this point, we have dx = 0, the tangent passing til rough it is parallel to the axis of abscissas, as in Fig. 13. The curve 2/ = a(l-cos^) gives dy^^a . X dx^ c c ' d^y a dx'' X -. cos - . c c d^y_ AVhen a? = } tc, then ^ = 0. When a? < J ;rc, then 3-~ > 0, and when xylite, then -r^ < 0. Hence at dx" ' ' dx^ x=:^7rc there is a point of inflection, whose coor- dinates are x =1^7:0 and y = a. The slope of the tangent through it is -5^ = - , and is a maximum. The curve has a minimum or- dinate {y = 0) at the origin, and a maximum ordi- nate {y = 2a) at the point where x = ttc. The curve (Fig. 14) extends indefinitely to the right and to the left of the origin ; for, making x= ± UTcCy the values will give a series of points of inflection, P,Q, H, . . . whilst the values INFINITESIMAL CALCULUS. 95 71 = 1, = 3, = 5, . . . will give a series of minimum ordinates {y = 0), and the values 71 — 2, =4, =6, . . . will give a series of equal maximum ordinates (y = 2a). Fig.X4 M A X The slopes of the tangents at the points of inflection will be alternately ax ^ c dx c 41. A cusp is a point at which two branches of a curve terminate, being tangential to each other. There are two species of cusps. In the first species the two branches lie on opposite sides of a tangent drawn at their common point; in the second species both branches lie upon the same side of the tan- gent. Thus, the curve of the equation y = b ±(x — a)i will have a cusp of the first species (Fig 15) ; for we find Fig. 15 dx and S = ±*(^-«)-*; dx ..\:r£>Ji:AL CALcrLVS. «M.i »ln>n -- /k «f lifiti ' dj' Tl..^ .•,.:-,(- >.;c^ ;!.,-:.,>- :w,. i.r*D(-bes. i* J/" and /\>. i!>r,M'. ,-v.i.»-.v. TitT .>:b;2 p.-aicsvf, downwards, \»I.;.li «).»<>: :;.ii^».! u..,y i.: :iif I'-dn! /*. whose oo- ■.Nis; !.s.,^i»r. / '., J = r^ tuiwUcb^i- not extend <«' » ■•• ). :; .1, 7", ;*r'.-3, i;:*r, »ii*-E J ^ a- 3/ is imagi- »' ...... ■ X.t »■ . iv.Tl. TirunclM* -wr 1. n- .- raunm be ■ ■ -T^-^ tiTstirli. INFINITESIMAL CALCULUS. 97 there is a cusp whenever the differential coefficient has two values which become identical at the point where the curve stops. 42. A multiple point is a point where two or more branches of a curve intersect, or touch each other. If the branches intersect, -J^ will have as cLx many values at that point as there are branches ; if tbey are tangent, these values will be equal. Thus in the curve we have dx" ^ ' '^ ^ Vl — af^ and when a?=0, then^ = ± 1, which shows that the curve has two branches intersecting at the origin ^^' (Pig. 17). As X cannot ex- ceed ± 1, the curve is limit- ed in both directions. The maximum and minimum or- dfj dinates correspond to ^ = 0, in which case we have of Vl-af = VI -of whence and the corresponding ordinate will then be y=±h 98. INFINITESIMAL CALCULUS. In the curve y=i±a? Vx'\-\ we have dy 50? + Ax dx~ 2 Vx+1 ' d/u When a? = 0, then 3^ = 0. Hence the origin O dx (Pig. 18) is a multiple point, where the two branches are tangent to each other and to the axis OX. The curve cuts the axis at the point rr =^ — 1 , where 3^= c» . ax Flg,18 In the curve we have X* + 2xl'y - y' = X whence = ±V-y±yVy + iy dy _ ± A Vy + 1 V — y ± y Vy + 1 dx ±3y±2-2Vy + l This expression, when y = 0, becomes either or = 0, according as we take the upper or the lower signs. But the real values of dy , , when 2/ = 0, can be found by putting the equation of the curve under the form which, when x = and y = 0, may be written thus «+^i-(2)=A INFINITESIMAL CALCULUS, 99 it being obvious that — ^ is nothing else than the ratio - , when these quantities are becoming each = 0. We thus find | = 0.andf=±*^.. These values show that one branch of the curve is tangent to the axis OX at the origin (Fig. ^^- ^^ 19), while a second branch, at the origin, makes with the same axis an angle whose tangent is = V% and a third branch, at the origin, makes with the same axis an angle whose tangent is = — V2. Hence the curve has a triple point at the origin, with two symmetrical loops under the axis of x. Its lowest ordinate is 2/ = — 1, corresponding to rc= ± 1, for which we have -^ = 0, aw Hence the curve, at the corresponding points M and iV", is parallel to the axis OX, 43. A conjugate point is a point whose co-ordi- nates satisfy the equation^of the curve, but which is isolated^ and therefore has no consecutive points. Thus the equation 2/' z=x^ {x — a) is satisfied by tlie co-ordinates x = 0, 2/ = 0, but no 100 INFINITESIMAL CALCULUS. Fig^tO other value of x less than a gives a point of the curve. Hence the point a? = 0, j/ = is an isolated point. The differential coefficient of the equation, which is dy _ 3g? — 2a dx~ 2Vx — a' shows that the curve (Fig. 20) has two branches cut- ting the axis OX perpen- dicularly at the point x = a. It shows also tliat when ir=0, ^ is imaginary ; which ex- presses the obvious fact that a solitary point can form no angle whatever with the axis of X. The second differential co- efficient of the equation, which is d^y_ 6 (a? - g) ~ 3a? + 2a dx = ± 4 V{x - ay shows that the curve has two symmetrical points of iniiection, P and Q, corresponding to the abscissa 4a x-=. — . 3 If a curve has a series of conjugate points, this series is called a hranche pointillee, or a dotted branch. For example, in the curve y^ = x sin* a?, every positive value of x will give two values for y; btit when x is negative, y becomes imaginary, ex- INFINITESIMAL CALCULUS. 101 cept when a? is a multiple of ;r, in which case y = 0. There will then be an indefinite series of conjugate points lying on the axis of x at equal distances, and forming a dotted hranch. As a second example, take the curve y = aa? ± Vx{l — cos x). Here for every positive value of x we have two val- ues of y, and consequently two points, except when cosii? = l, in which case the two points reduce to one. Hence the curve, owing to the periodic co- incidence of such points, will consist of a series of loops not unlike the links of a chain, having for a diametral curve the parabola y = ax*. But for all negative values of x the ordinate is i maginary , except when cos x=li \ and in these cases we have a seiies of isolated points situated on the left branch of the diametral curve. Order of Coniact, OsculcUion. 44. Let y = = 1 -m' On the other hand ds hence = f/M+lXf = . On the other hand, y ^MH=PN ^ON -OP :=zr — T COS ^. Hence cos 1? = ^, sin *=/|/l - (?L3i/) =1 ^2?:^^^', * = cos-i ^^^^ . r Substituting these values in the expression for x^ we have xz=zr cos"^ — v'2r2/ - y", which is the equation of the ascending portion of the cycloid. From this equation we obtain dy _ V^y-y" _ ./2r_. dx- y yy '' whence dx^ y ^ dx dx y* ' 114 INFINITESIMAL CALCULUS. ^{^y Vj a) + daF{x, y, a) = 0, a alone being supposed to vary. And since we Lave already the term .F{x, y, a) =0, we shall have also da F{x^ y, a) = ; and the solution of the case will wholly depend on these two equations. The point of intersection given by these equations changes its position for every different value given to a, and will describe a continuous line when a varies without interruption. If, then, we eliminate a between the same equations, the resulting equa- tion if (x^ y) = will be the equation of the curve formed by the successive interaection of all the curves derived from the equation F {x, y, a) = 0, when a is supposed to vary continuously. The curve

whence a*y = 4m' (a — x). Differentiating this equation with regard to a, we find a = — ; and this value substituted in our y equation gives xy = vC ; and therefore the curve is a hyperbola referred to its asymptotes. 50. The above theory may also be applied to the solution of problems in which the pammeter de- pends on the variation of two quantities. In this case, the data of the problem must give rise to two equations. Let us give an example. The straight line PQ (Fig. 26) slides between ^ two rectangular axes. Find the envelope^ or the locus of its inter sec- q tions. For the solution, let AX, J Fbe the co-ordi- nate axes. Let AP = a, AQ = b, PQ = c. Then the equation of the line a PQ will be Fig.te which, for greater convenience, may be written thus: INFINITESIMAL CALCULUS, 117 and in which a and & are subject to the condition a' + 6* = c\ (2) Differentiating (1) and (2) with regard to a and 6, we obtain ^ + 2^ = 0, ada + Ml=0, whence, by eliminating da and 6^, we shall find X y X a^y , y Vx ~ = f5, or -=-Tr, andx^ = — r. Substituting, in succession, these values in (1), we obtain y{a' + V) = V and x{a' + V) = a\ or yc" = &•, and x(f = a* / whence a = ojic*, 6 = j/ici Substituting these values of a and 5 in equation (1), and reducing, we finally obtain the equation of the envelope, which is art -f- yl = c*. Elements of Arcs. Surfaces y and Volumes. 51. We have seen (No. 33) that if ds is an in- finitesimal element of the line s referred to rectan- ' gular axes, we have ds = Vdaf + dy\ 118 INFINITESIMAL CALCULUS. the values of dxvmdi dy being drawn from the equa- tion of the line. This equation gives the differen- tial of any arc. The differential of an area is the infinitesimal area comprised between two con- secutive ordinates of the curve. Thus, if AP and BQ (Fig. 27) are two con- secutive ordinates, the area APQB will be the differen- tial of the area comprised between the curve and the ^ axis OX. Now APQB^ ydx-\-\dydx. Hence, denoting APQB by dA^ and neglecting the term i dydx^ we shall have dA = ydx as the expression for the differential of a plane area. If the element PQ of the curve be made to revolve about tlie axis OX, it will generate an infinitesimal element of a conical surface. Denote it by dS. Its expression will be d8^ 4 12;ry -f 2;: (?/ + dy)\ Vdx' + dy^, or, reducing. dS = 27:y Vda^ + dy" ; and this is the differential of a surface of revolu- tion. If the area APQB be made to revolve about the axis OXy it will generate an infinitesimal element of a conical volume. Call it d V. Then dV= * 1 ^y" + ^{y + dyy + 7:y{y + dy) \ dx, INFINITESIMAL CALCULUS. 119 or, reducing, and this is the differential of a volume of revohUion. Bifferenbials with Polar Co-ordinates. 62. Let p=zf{f) be the polar equation of any curve PL (Fig. 28), the pole being at 0, and OX being the initial line. At the point P we have p = OP^ and ^ = POX. Draw the tangent PB, the normal PC^ and through the pole draw BO perpendicular to the radius vector. Then OB will be the subtangent^ and OC the svbnormal^ while OA perpendicular to the tan- gent will be the polar distance of the tangent Let ds = PQ be an infinitesimal element of the curve. Draw the radius vector OQ^ and, with as 120 INFINITESIMAL CALCULUS. centre, the arc PR. The infinitesimal triangle PQR may be considered rectilinear, and right-angled at R. Hence P^ = PR + RQ\ But PQ = ds^ PR = pdip^ RQ =dp ; hence, substi- tuting and extracting the root, ds = Vdp' + p'df. Such is the differential of the arc. Let dA be the differential of the area swept over by the radius vector. The infinitesimal area POQ may be considered as the area of a rectilinear tri- angle having the base OQ and the altitude PR. Hence dA^\OQy.PR^\{p-\'dp)pdip, or, as dp disappears by the side of />, dA = \ p^dif. Such is the differential of the area. Let F denote the angle OPB of the tangent with the radius vector. This angle and the angle RQP^ may be considered equal, as they differ only infini- tesimally. Now therefore '•■'^«^=i=^'- ten V=«' dp Let OA-p. We have OA-OP sin OP A; therefore . ^ ta n Y j> = /,smT; or ^ = ,^===. _^-. INFINITESIMAL CALCULUS, 121 Substituting the value of tan T, and reducing, we have p*d(p Such is the polar distance of the tangent. Let S, 7^ denote the subtangent. We have >». r=05= OP. tan y, or S.T^p' dp Let /S'.iV denote the subnonnal. The proportion OC: OP:: OP: OB gives ~" "~ OB Q^dtp^ dip Let PC^R (Pig. 29) be the radius of curvature of the line PL at the point P. Drawing OM perpen- dicular to PC pro- duced, and joining the points and (7, we have from the triangle POC dp Fig.f9 OC'^OP'+PC" --^OP.PCcosOPC. \ But OP cos OPC = PM=z OA =i?. Therefore 00'-p' + Jr-2pIl. 122 INFINITESIMAL CALCULUS. Now, if we pass from P to Q^ the side OC and the radius M will remain unchanged, p and p alone being subject to variation. Differentiating, then, and considering 06^ and R as constants, we lind = 2pdp — 2Rdp^ whence R = ^-^ • Spirals. 53. A spiral is a plane curve generated by a point moving on a straight line while this straight line is uniformly revolving about a fixed point or pole. The portion of the curve generated during one revolution is called a spire. The law according to which the moving point advances along the re- volving line determines the nature of the spiral. Denoting by p the radius vector, and by f the angle that the radius vector makes with the initial line, and considering /> as a function of f , the general equation of a spiral will be The most remarkable spirals are the spiral of Archimedes^ the parabolic spiral^ the hyperbolic spiral^ and the logarithmic spiral. 64. The spiral of Archimedes is generated by a point moving uniformly along a straight line uni- formly revolving. If p and p' be two radii vectores, and tp and ' = 2 t:. INFINITESIMAL CALCULUS. 123 Tip T Then p = ^^ or, making for greater simplicity 5- = a. 27t p = a(p. Stt (1) Sucli is the equation of the spiral of Archimedes. Its form is the same as that of the equation y = dx of a straight line passing through the origin of co-ordinates. Tills spiral may be con- structed by dividing the cir- cumference into a number of equal parts, and the radius into the same number of equal parts, and then taking as many parts of r for the radius vector as there are corresponding parts taken on the circumference. Differentiating (1) we have dp = adif Substituting this in the general formulas of No. 52, we shall find ds = adip Vl + f •, dA = ^cCif^dif^ tanF= ^ =fl>, p= — ^ Vl + y' 72 = ^(1+?!)', 2 + 9^" Fig, so 124 INFINITESIMAL CALCULUS. It will be remarked that the subtangent CB cor- responding to f = 2;r is equal to the circumference of the measuring circle ; for ay" becomes a(2r)" = jj^ (2;r)* = 2;rr. Hence the area of the triangle ABC is ir X 2;rr = ;rr* = area of the circle. 55. The parabolic spiral (Pig. 31) is so called be- cause its equation />• = 2af (2) is of the same form as the equation y* = 2ax of the parabola. It may be con- structed by first constructing ! the parabola and then laying oflf from A to D along the / circumference of a measnr- / ing circle any assumed ab- I scissa Xy and drawing from \ the centre C towards D \ the corresponding ordinate y as a radius vector. Let r be the radius of the measuring circle ; then - be- comes y, and y becomes p. Hence after a revolution we shall have - = 2;r, and y = CA = r, and the equation (2) will give r"= 2a X 2n-, or a = r-. Diflferentiating (2), we have pdp = ad = 00 ; which means that the curve begins at an infinite distance from the pole. On the other hand, p cannot become = unless ip becomes infin- ite ; and this shows that the carve can make an in finite number of spires in approaching the pole. L 126 INFINITESIMAL CALCULUS. Again, when f = 0, the curve and its tangent are parallel to the initial line GX^ and the equation of the polar distance of the tangent gives p = a. Erecting at C (Fig. 82) CD = a, peri)endicular to CX^ and drawing DR parallel to CX, the line DM, if prolonged to infinity, would be the tangent in question. The equation tan V=: —

, (4) in which

, the arith- metical progression ip "iip Zip U, - 5 > a a a , . . . will entail a geometric progression Id a Ot for the corresponding radii vectores. Hence, if we 1 28 INFINITESIMAL CALCULUS. describe a circle with the radius > , and divide its circumference into equal parts, and if then from its centre we draw through each point of division in- definite straight lines, and on these lines we take such lengths as are required in order to form a series of radii vectores in geometrical progression Vthe ratio being e /, we shall determine any num- ber of i)Oints belonging to the spiraL PART II. INTEGRAL CALCULUS. 68. The object of the integral calculus is to ex- plain how to pass from given differentials to the functions from which they may be derived by dif- ferentiation. The functions thus obtained are called integrals^ and the operation by which they are found is called integration. To express that a function x is the integral of dx^ we write fi and the sign/ prefixed to the differential is called the integral sign. Integration and differentiation are inverse opera- tions. It follows, first, that, since the differential of a polynomial is the algebraic sum of the differ- entials of its terms, the integral of a differential polynomial must be the algebraic sum of the integ- rals of all its terms. Secondly, since constant factors are not subject to differentiation, they are not subject to integra- tion, and may, therefore, be written without the sign// thus, if a is a constant, tadx = a Idx = ax. Thirdly, since a constant term disappears by dif- ferentiation, a constant must be added to the in- tegral obtained, to make it complete. We deter- 180 130 INFINITESIMAL CALCULUS. mine the value of such a constant, in each particular case, by making it agree with the conditions of the problem proposed ; but, as it is capable of being determined by various arbitrary conditions, it is commonly considered as an arbitrary constant. Before any value has been assigned to the con- stant the integral is said to be indefinite; when the value of the constant has been determined so as to satisfy a particular hypothesis, the integral is said to be particular ; and when, moreover, a definite value has been assigned to the variable, the integi-al is said to be definite. 69. The following example will show how defin- ite integrals can be obtained. Let ^dx be the function to be integiTited. We already know that d {x*) = 3x*dx. We have therefore /safdx = X* ; and this is the incomplete integral. Add to it a constant C\ then r^dx:=zaf-\^C; and this is a complete integral ; but it is still in- definite. Assume now that the particular problem under consideration requires the integi-al to be = when ir = a, that is, that our integml begins at x=.a. In this case we shall have and consequently (7= — a*. Hence we shall write I ar'eto = a;' — a", INFINITESIMAL CALCULUS. 131 the letter a placed at the bottom of the sign / in- dicating the inferior limit of the integral, viz., the place where it begins ; and the letter x at the top of the same sign indicating tha superior limit where the integral ends. But, as re is still vari- able, this is a particular integral. If our problem requires the integral to end at a; = ft, we shall have at last %J a and this will be the definite integral. In geneml, if we have ff{x)dx = F{x) + C, assigning to the integral the limits required by the conditions of the case, say a and ft, the definite in- tegral will be expressed by rf{x)dx = F{b)^F{a). J a The present treatise will contain three sections. In the first we shall explain the various methods of integration; in the second we shall apply the in- tegral calculus to many questions of Greometry; and in the third we shall solve by it various mechanical problems. 132 INFINITESIMAL CALCULUS. SECTION L VARIOUS METHODS FOR FINDING IN- TEGRALS. 60. Before we proceed further, let the student be warned that, although in the differential cal- culus we can always pass, by a uniform method, from a given function to its differential, in the in- tegral calculus we have no general method for pass- ing in all cases from a given differential to its integral; and for this reason we are not unfre- quently compelled to make use of devices of various kinds, the choice and employment of which is not always exempt from difficulties. We shall start from the fundamental formulas which regard the simplest cases of integration; after which we shall explain various processes used for the reduction of other less simple cases to forms integrable by the same fundamental formulas. Integration of Elementary Forms. 61. The fundamental formulas of the integral calculus are found without labor by simply revers- ing the corresponding formulas obtained in the differential calculus. Thus, since the expression , ^ has aaf^dx for INFINITESIMAL CALCULUS. 133 its differential, we immediately see that the integral of dof^dx is — rr • Therefore 71 + 1 fa^^=^^^a (1) This formula is general, whether n be positive or negative, integer or fractional. The only case to be excepted is ti = — 1, because in this case — r-r- becomes - , and gives no differential. On the other hand, ax'^dx is not the differential of a power, but of a logarithmic function; for dx ax'^dx = a — = a^ log x ; and thus fax'^dx = ^ /— = a log 0? + (7. (2) The expression a* log adx is the differential of a*/ hence /^a.^ = _^ + a (3) y loga ' By reversing the differential trigonometric for- mulas of No. 17, we shall find the following : /cos xdx = sin a; + C. (4) /sin xdx = — cos 0? + (7. (6) /l^=taii^+a («) y cos* X dx y sin* a; (7) 134 INFINITESIMAL CALCULUS, /sin ocdx = vers x-\- C. • (8) /cos a^dx = — covers x-\- C. (9) /sin a;<^ , ^ ,^^,v r — = sec a: + C. (10) cos a; * ^ ' — ^i — = — cosec x4' C. (11) sm X ' ^ In like manner, if we reverse the differential formulas of No. 18, we shall find the following : yj^ = tan-»2/+a (14) /l^ = - cot-» y + a (16) /■;^=^ =-«"-• 2^+^- (i«) f^^x-^-'y+<'' (i«) J^ In these latter formulas, make v = — ; then INFINITESIMAL CALCULUS, 135 dy ^- dx; and by substitution and reduction we shall obtain the foUoiwing: A-7-^-^=--i<»8-'^ + ^- (21) J Va' — b*af a ^ ^ ^ rdg^^ = 1 ers-' ^ + C (24) f , ^ — =-\ covers-* - + C. (25) Z ' , ^^ = 1 sec-» - + a ^26) /-7^fr= = -lcosec-'*^ + e. (27) By these formulas a great number of differentials can be integrated. We give here a few examples for the exercise of the student. EXAMPLES. 1. dy=ixdx^ yz=i\a?. 2. dy = —z=-^ y^2aVx. 3. dy^zoafdXj y^^jOLx^. J J -I '' 136 INFiyiTESIMAl, CALCULUS. 4. dp = 2a:^dXj y = +^*5 6. dt/= — ix'idx^ y = J^-i, 6. dy^=^x'n dx. y = ic » . 71 — m 7. dy = 2x'^dXy y = log{af). 8. tZy = ( 2ir j €?ir, y:=af- log a?. 9. dy =a'^' cos xdXj y = ^Btoar log a ' 10. dy = 2 cos (2a:) rfa;, y = sin 2x. ^^■^y = TJv. 8^=8 tans- ec-g) 2 12. ^2/ = si^ (^ ^> y = - vers {ax). a? y = — covers ^ • 13. (^2/==^ c^sf^jrfir, 14 dv ^^ y = i sin-^ Sa?, "• "^^ f'l-4a:" 1R ^,,_ 2a^ y — cos-^(a:^. ^"- ''^ 4/1 -a;*' 16. ^y 1+^- 2/ — itan-^(a^. - ^ - «- - • INFINITESIMAL CALCULUS. 137 sin ^7(2^ ^^•^^^^T^l^f^?^' y = 4COS-' (2 cos a:). COS OCCl'OR ^- ^^^l + sm^' y = tan-^sin ir). •^ 1 + a**^ ^ log a In the above examples we have omitted the arbi- trary constant, as it had no bearing on the work proposed. 62. When the fundamental formulas above given are used for the successive integration of differ- entials of the second, third, or any higher order, a constant is to be added at each integration. Thus, if the expression ^ = ax is to be integrat- ed, we first multiply both its members by dXy which gives us -3-^ = aa>dx. or rather d 3-^ = axdx. ax ax 138 INFINITESIMAL CALCULUS. Integrating this, and adding a constant, we have ^ - T + ^• Multiplying this first integral by dx, we have and integrating, and adding a new constant, do? 6 ' ' Multiplying this second integral by dx^ integrating, and adding a third constant, we shall at last obtain It is a general rule that every complete integral must contain as many arbitmry constants as there have been successive integrations performed for obtaining it. Each constant is, of course, to be de- termined by taking the integral between the limits required by the particular conditions of the problem to be solved. Reduction of Differentixils to an ELemerdary Form. 63. When the proposed differential is not in the form required for integration, it may often be re- duced to a proper form by some simple algebraic process. Thus the differential INFINITESIMAL CALCULUS. 139 - will be reduced to an integrable form by perform- ing the operation indicated by the exponent of the parenthesis. We shall then have Thus also the differential will be properly w "~" fl/ reduced by division. As = —{ a? -\- ax -\- a^ ]dx. a — x V a - 0?/ ' hence /X^CbX X CLX • % t / \ If we had a differential of the form {a 4- bx) dx a'+af ' we might split it into its parts by writing adx bxdx whence mf^ = 'a- f + * log v^^. a' + a? In like manner the differential xdx V2ax — X* will be reduced to two integrable terms by simply 140 IXFIXITESIMAL CALCULUS. m _ ■__■ _____■_■■ adding and subtracting the quantity a in its nume- rator. We obtain thus xdx adx _ {a'-x)dx V2ax — of V2ax — a? V2ax — of ' whence xdx _, X V2ax-'X^ The differential / . _^ = a vers-* ^ — V2ax^ of. dx Vaf — a* ■ I 11^ III! X may be reduced to an integrable form by multiply- tag iU nomentlor and iu d>>minator by ^wH. We thus obtain dxV3^^-a^ X a? — cC _ ajeZic a'^ (IX^=- whence /i^Llf! rf^ = V^^' - a sec- ? . J X a 64. Sometimes a differential is reduced to an in- tegrable form by the introduction of an auxiUary variable. Let dy = a? Va + xdx. If we make a-^x^^z^ whence dx = dz^ a;' = (a — a)', we obtain dyz=:{z — ay zi dz, INFINITESIMAL CALCULUS. 141 which, being developed, gives and or, replacing by a + a?, and reducing. y 4a , . . . 2a' = JK« + ^y-|'(a + ^) + ^[ i^FT^- Again, let it be required to integrate the differen- tial , dx Assume of ±a* = 2:', whence xdx = zdz. Adding sdx to both members of this last equation, and factoring, we have {x-[-z)dx = z {dx + dz), whence dx _ dx-\-dz __ and, integrating by formula (30), f {a + baff af'^ dx = or, making, for greater simplicity, a + baf^^X, and consequently r X'ar-^ dx = Transposing the last term to the first member, and reducing, we get h (np + m) fY^^-ta-r- nl>{p + l)J^ ^- X"'^ af-" _ a (m-n) r y.^ « « i ^ w6CP + 1) nb{p + l)J ^ ^ ^'^' INFINITESIMAL CALCULUS, 155 and dividing by the coefficient of the first member, T? 1 — ^ — TT — \ — \ / X^ af'""^ dx. (33) By this formula the integral of X^ af'^ dx is made to depend on that of X* x'^~'^'^ dx^ where the exponent of the variable is diminislied by n units at each application. The formula fails when Tip + w = ; but then the integration can be made by formula (32). Second case : To lower the exponent p of the bi- nomial. We have identically (a + baf'Y = a (a + bafy-^ '{-baf{a + baf'Y'\ or X^^aX^-^ + baf' X^ \ whence fx^ af-^ dx^afx^'"^ af"^ dx+b fx^ ^ ^"'^•*-^ dx. Now, the last term of this equation can be reduced by formula (33) ; for, chianging m into m-^-n^ and p into i> — 1, and multiplying by 6, that formula gives b fx^'^ ar^""^ dx=i X^ of am /j^p-i ^-1 ^ np -\-'Vi np-\- in Substituting this for the last term of the above 156 INFISITE8IMAL CALCULUS, eqoation, uuitiiig the similar terms, and reducing, we obtain /X'af*-*£te=— 1^+— r^-A'''^~"'^^- (34) J np-^-m * np+mj ^ ^ By this formula the inte^in^l of X* af^'^ dx is made to depend on that of X^~^ af'^ dx^ where the exponent of the binomial is diminished by unity at each application. When Tip + 7» = 0, this for- mula fails ; but then the integration can be made by formula (32), as already remarked. Third cdse: To increase the ea^ponent of the THiriable, When m is negative we Inay need to diminish it arithmetically or increase it algebraically. To do this we ' proceed as follows. Reversing formula (33), we obtain Xp a-«»-»-i dx = This equation, by changing m into — m + ti, will become am ' am J ^ By this formula the integral of X* rr""*"^ dx is made to depend on that of X^ ^-w+^-i dx^ where the exponent of the variable is increased by n units at each application. Fourth case : To increase the exponent of the binomial. INFINITESIMAL CALCULUS. 157 When p is negative, and we need to increase it, we proceed as follows. Reversing formula (34), we obtain J anp ' anp J This equation, by changing p into — i? + 1, will become /x-* af'^ dx = X'P^' or m'\'n-'np C^^p.x ^., ^ /ogx a/i(^-l) an{p^l)J^ ^ ^' ^^^^ By this formula the integral of X"^ af^~^ dx is made to depend on that of X"^*^ af~^ dx^ where the exponent of the binomial is increased by unity at each application. YS, The integration of the expression af^dx V2rx — X* can be made to depend on that of a like expression, in which the exponent g is diminished by unity. We have identically ^ofi dx {2rx - af)'^={2r-xy* a^-* dx. V2rx — of Now, representing (2r—a;)-i by X"i, and comparing with formula (33), we have a = 2r, &=-l, n = lj p = — i, m = y + J; whence 158 INFIMTESIMAL CALCULUS, b (np + w) = — g, a{m — n) = r{2q — 1), i? + 1 = i, 77i — 7i = g — J. Substituting these values in (33), we find / X-iafi-idx= q ^ q J ' that is, f V2rx — a? ofi'^ V2rx — a? r (2g — 1) r af^-^ dx , 2 i J V2rx-af '' ^ If g be entire and positive, by a successive appli- cation of this formula, we ultimately arrive at the form / dx . X = vers ^ - . i^2rx — of In like manner the integration of the expression a^ dx V2rx + x' can be made to depend on tliat of a like expression i in whicli the exponent q is diminished by unity. } For the only difference between this case and the preceding one is that b is now positive instead of negative, and therefore b {np-^m) = q ; which shows that the signs of the second member of the formula ought to be changed. Accordingly INFINITESIMAL CALCULUS, 159 / of dx V2rx + a?~ af^'W2rx -\- of __ r (2g - 1) r afl'^ dx . Q Q J V2rx+af' ^ ^ Formulas (33), (34), (35), (36), (37), (38) are called formulas of reduction^ and have a wide range of application. EXAMPLES. 1. To integrate x'dx ^y^-TTr. Here we have X= r' — a;', a =r', 6 = — 1, ti = 2, jp = — i, m = 3. By formula (33) we have, for this case, fx^ of dx^ ^^ - zVy ^"* ^' that is, or /of dx X .-- , r' . . x 2. To integrate dy^^afdx Vr^ — of. Here we have X= r^ — a?^ a = r\ J = — 1, 7^ = 2, ^ = J, m = 3. 160 IXFIXITESniAL CALCULUS. Formnla (33) gives, in this case, fx\ of dx=^ - ^\f^^ ^^ that is, To integi'ate the last term of this equation we use formula (34), in which we shall make X= r' — aj", a=^r^^ ti = 2,j? = i, m = 1. We obtain fx^dx^^ + '^fx^dx, that is, Jdx'fF^'a?^^ i^r* - 2^ + 1 sin-* ^. Substituting this in the preceding equation, we have J a? dx Vr^ - of = ~f Vif^^'V^ i/p=r^+g sin-* f • 3. To integrate dx dy = Here we have X= 1 + 55*, a = 1, 71 = 2, —p = — 2, m = 1. INFINITESIMAL CALCULUS. 161 By formula (36) we have that is, or Integration of some Trinomial Differentials. 74. A trinomial diflferential of the form — {a-\'bx±off xT dx m can be made rational in terms of an auxiliary vari- able, when p and m are whole numbers. When p is even the expression is already rational ; when jp is odd, then, making jp = 27i + 1, the expression be- comes {a + bx±afy' {a + bx±of)i af^dXy in which the only irrational part is Va + bx ± of. When x" is positive we obtain a rational form by assuming Va-\'bx-{'af =zz — X, from which we shall get a-\-bx = z* — ^zx^ 162 INFINITESIMAL CALCULUS. and Va + ftaj-f^ = — 2z4-h — * Thus the given expression will become rational in tenns of z^ and therefore integrable. After the in- tegration it only remains to substitute for s its value When a:' is negative let us assume Va + bx — af=: V — {x — Ti){x ^Jc)^{X'-'h) z^ - where h and k are the roots of the equation a:* — &aj — a = 0. We then have a + *^ — ^ = (^ - A) (A — ic) = (a; - A)" ^"5 or Hence and , _ 2{Jc — h) zdz ^- (1 + zy i^a+bx-af = LJ/ " ^) '^~ 1 {k - h) z Thus the given expression will become rational and integrable in terms of z. After the integration it will only remain to substitute for z its value \ x--Ji INFINITESIMAL CALCULUS. 163 EXERCISES. cLiR 1. dy = ^ , y = log(ar+l+2 Vl+x+a^. dx , /l — X x + 2' Integration by Series. 76. When an expression Xdx^ in which X is a function of Xy is to be integrated, it is often con- venient and useful to develop JTinto a series by any of the known methods, and then to integrate each term separately. This is called integration hy series. If the series obtained be convergent for any particular value of x^ we shall obtain the ap- proximate value of the integral for that particular value of X. Thus, given J dx ^^2^ = 1+^' we may, either by the binomial formula or by mere division, obtain 1 Whence dx — X "^ X "T~ •!/ ^~ •!/ ~y" X ~"" • • • = dx — xdx ^- a?dx — sfdx -f- x*dx — . . • 1 + a: and CLX XT , •€/ X , XT T wUclx is the expression for log (1 + oS). /dx _ a? .a? a? .(& 164 INFISITESIMAL CALCULUS. Again, given , dx we may, by simple division, obtain the series 1 whence — X ^~ SD "J" vu *~" Su "^ uT ""~ • • • y • • • ^ , , zzzdx — oifdx + x^dx — afdx + ^dx — and / dx _ - _^i^_^i5^_ 1+^ "3"'"6 7"^^9 which is the expression for tan"* x. In a similar manner, if we have dx we may obtain, by the binomial formula, ^ . Cu I ox . o»ox . o»OmiCcr . whence, multiplying by e^a?, and integrating, we find dx /dx ^"^2.8 "^2. 4.5 "^2. 4. 6.7 + 2. 4.6,8.9"^ • • ' which is the expression for sin*^ x. INFINITESIMAL CALCULUS. 165 76. Let us have to integrate the general expres- sion dy = Xdx^ where X is a function of x. Integrating by parts, we have y = Xx — fxdX=. Xx — / -^ . xdx. This last term, again integi'ated by parts, will give J dx ' ~ 2 dx J 2 ' dx ' This last term will give, by the same method of in- tegration, ^ d^X_ 2 ' dx ~ a? ^ d^x X* d'x r x' qx da? ' f 2 dx^ ""2.3 da? J 2.3' and continuing to integrate in the same manner, we shall find y :=fXdx=z dx '^2 "^ da? ' 2.3 dx' ' 2.3.4 + • • • . This elegant formula, by which the integral is obtained through successive differentiations, was discovered by John Bernouilli, and bears his name. Integration of Trigt)no7netric Bxpressions. 77. Trigonometric expressions can be reduced to integrable forms by suitable transformations based on the correlation of trigonometric lines. 166 INFINITESIMAL CALCULUS. dec 1. Given dy=.- — , we have from trigonometry sin 0?= 2 sin \x cos jta?=^2 tan \x cos' \x; whence dx dx sin a; "" 2 tan ^x cos* \x ~ • 1 ^dx _ d (tan \x^ tan ^x cos" \x ~ tan Ja? and therefore 2/=/s|^ = log(tan4a^). . (89) dx 2. Given ^2^ = cos X ' we have cos x = sin (90° — rr). Hence, substituting 90° — X for 2? in (39), we shall have 3. Given y = r , ^ tan a? we have dx _ cos g?^rg _ d (sin a?) . tan a? "" sin a? sin a; * and therefore 2/ ^a? =/tan^ = ^°S ^^^'^ ^)- (*1) 4. Given y = — r— , ^ cot a? ' mFINITESIMAL CALCULUS, 167 we have dx _ sin xdx d (cos x) , cot X " cos X cos X ' , whence 5. Given , dx ^ sin X cos a? ' we have sin X cos a? = i sin 2x. Substituting, and integrating, /dx r 2dx , ,^ . = / -; — TT = log (tan x). sin a? cos a: y sm 2a? ® ^ ^ 6. Given eZy = sin"* X cos* a:^, it will be convenient to transform this differential into an algebraic expression by assuming sin x=^z^ w^ience cos x=z V\ — 2', and dz dx=^ Thus we shall obtain the differential in the form n-l e?2^ = (1 — z") ^ s"* dz^ which can always be integrated when w + 7i is a whole and even number (No. 71). If m + 71 is a whole but uneven number, then the integration may be made by formula (33) or 168 INFINITESIMAL CALCULUS. (34), by which the exponents will be graaually reduced till we reach some simple and elementary form. Formula (33) will give us or / sin"* X cos* ajtf^ = sin"*"* X cos*"^* re . m — 1 H i — /sin"*'* a? cos* xdx : from which, if ti = 0, we obtain / sin"* xdx = sin"*"* X cos X . m — 1 m Formula (34) will give us H ^in"*"* xdx. /(I - ^) ^ ^ dz n-l »-8 ^^*(l-^')« ,71-1 /*,, ^,V m^ m+n ' m + nj ^ ' ' or ^in"* X cos* xdx = sin**"^* a? cos*"* a? . 7^ — 1 I 7» + 71 ' m + 71 A ; — / sin"* X cos*"* xdx; ' m -\-nJ INFINITESIMAL CALCULUS, 169 from which, if m = 0, we obtain y^ - jj sin X cos*'* X . 71—1 r - « ^ cos* icao: = /cos* ' a:aa:. Integration of Logarithmic Differentials. % 78. Let (iy = af^'^ (log a?)* rfa? be proposed for integration. Assuming (log a?)* =s i^, and af^'^ dx=^dVy we obtain du = ^ ^ ^ — tto, and t) =— • X m Hence, by formula (30), faf"^ (log ir)* ^oj = J (log 0^)* ^ ^y^"^ (log xf' dx. (43) By this formula we can reduce the exponent of log a: by 1 at each application. The formula fails for 7» = ; but then the integration is obvious. If we reverse (43), we obtain / af* * (log a?)*"^ dx := — (loga?)» - ^y"^'^"* (log a?)* dx, and, if we change ti — 1 into — ti, we shall have / af^"^ dx (log a?)* — ""(7i-l)(loga?)*-* + 7i-iy (log a:)*-*' ^^ and by this formula we are enabled to reduce the exponent of the denominator by 1 at each ap- 170 INFINITESIMAL CALCULUS, plication. The formula fails when ti = 1 ; but when m = and n = 1, the integral becomes X lofi: X """ loK ;z? "" o ^ o > ir log a? dx To integrate d^ = , — ~ ^ it suffices to make log x (f dz =iZj whence 0? = ^*, dx = e* dz ; and dy=^- — -- z And, as^=l + e + o"f-o~Q+ • • • ' ^^ \w\^- gration is obvious. Integration of Exjmnential Differentials. 79. Ijet dy^^af" a"" dx be proposed for integration. Assuming a?"* = 2^, a* ^a? = t?i?, we obtain du = mcc^ ^ dx, v = log a ' and formula (30) will give /af^a'^dxzzi 1 — - — i~^ /af-^ a^ ^^r. (4fi) log a log a y ^ '^ At each application of this formula the exponent of X will be lowered by 1, till we reach the simple form >■ dx. When m is negative, by reversing the formula we first obtain yaf"'^ a* dx= ^^ /of a* dx, then, replacing m — 1 by — 77^, and reducing, / a"" dx _ a"" , logg P a'^dx ..g> • INFINITESIMAL CALCULUS. 17 1 By this formula the exponent of the denomina- tor is lowered by 1 at each application. But the formula fails wlien m = 1 ; and in this case the in tegral of is obtained by changing a* into its CD development X* a? l+(loga)a;+(loga)' 2+(loga)'g^+ . . . 80. Let dy = Xa* dx, X being a function of x. Integrating by parts, we liave ■ fXa' dx = f^ - fr^^ dX. J log a J log a If we take the successive diflferentials of X, and place dX=X' dx, dX' — X"dx, dX'=X"' dx, . . . we obtain J log a (log a)" J (log a)' r «' dX' - ^'^^ _ f "' dX' J (log a)' '*"*■ - (log of J (log ay ^ ' and so on. Hence, by successive siibstitutions, we shall find f Xa''dx = ( ^_ _ X' , X' _ X"" ) I log a (log a)' "^ (log a)' • • • ± ^log «)»♦* f raf^dX^ ■ +y (log «)•*»• ^ '^ a* •^ 172 INFINITESIMAL CALCULUS. If X is such a f auction that one of its differential coefficients X\ X^, ... is constant, then the next differential coefficient will be =0, and the 7| ;—j will vanish. In such a case the integral will be exact. Thus, if we have then X=af-/i\ dX=2xdx, X' = 2x, dX'=2dx, X" =2, dX''=0. On the other hand, log € = 1. Substituting in (47), we find p = €f{af-h' ''2X + 2). Integration of Total Diffei'entials of the First Order. 81. The total differential of a function i^=/(ir, y) is, as we have seen (No. 19), , du J , du J in which -j- dx and -j- dy are the partial differen- dx dy ^ ^ tials of the function. We have also seen (No. 25, Scholium) that the existence of an- exact total dif- ferential of such a function entails the existence of the relation lis ^ 1® . dy dx ' whence it follows that a differential of the form dn = Mdx + Ndy INFINITESIMAL CALCULUS. 173 will be an exact total differential when M and N are such as will satisfy the condition -j- =-:t— ; •^ dy dx for, in such a case, we evidently have M-=--^ , and nr du dp If the condition is satisfied we may immediately integrate the partial differential Mdx and write u = fMdx+ r, F exhibiting a function of y which is to be deter- du mined so as to satisfy the condition -3- = iT. Ac- cordingly, we now differentiate our integral with respect to y, and, dividing by dy, we have du _ ^f^dx ar^ ^ whence and dy dy dy dy~ dy ' ^ r i dfMdx\ ^ and finally u= I MdxAr j I N- ^l~ | dy. (48) 1 7 4 INFINITESIMAL CALCULUS. EXAMPLE. Let du = {^axy — ^bafy) dx + (flw;' — ha?) dy. Here we have whence -J— = 2ax — Sbx* = -5— . dy dx Hence our total differential is exact and unmedi- ately integrable. Integrating the partial differen- tial. iH/ir/rc, we have u = f{2axy — Sbx^y) dx-}- Y= aa?y — hafy + T. Differentiating this integi'al with respect to y, we find du , ^ ^ , dY ^^ dy ^ dy ' whepce ^=N- {ax' - bx') = 0, and ¥=0. And therefore u = aa^V — bx*y + ^« EXERCISES. 1. e^t^ = Sx'y'dx + 2x*ydy^ u = x^y* + ^- 2. ^t. = ^ + (2.y-|)^y, t,=| + j/ + 0: y -^ar ^ x INFINITESIMAL CALCULUS, 175 Integration of the Equation Mdx + Ndy =0. 83. The implicit function / (rr, y)=0 has for its diflferential (No. 21) where ^ and ^ are the differential coefficients of ax ay the function taken with respect to x and y. Repre- senting -^ by M^ and ^ by iV, the differential will take the form M(fx + Ndy = 0. Now, this equation, whenever we have ~— = -j- » will be an exact differential (No. 81), and may be integrated by formula (48) ; but, as we have here du = 0, the integral will be n=^ C, When by any transformation the equation can be placed under the form Xdx+ Ydy=iO, X being a function of x alone, and Za function of 7/ alone, the integral can be found by taking the sum of the integrals of the two terms. Thus fxdx+f Tdy= C. When the equation can be placed under the form Ydx + Xdy = 0, or under the form Xrdx + X'Y'dy = Oy 176 INFINITESIMAL CALCULUS. the variables can be separated by division. Thus y + ^=0, and j^,dx-\-Y ^^ = 0, and when the variables are thns separated the in- tegration becomes possible. Hence the separation of the variables has been one of the main objects of study on the part of mathematicians. EXAMPLES. 1. Given ydx — xdy = 0. Divide by xy. Then — - -^ = 0j log x-\ogy — C^ log c; and therefore •27 -=c, or x-=zcy. y " 2. Given ity'dx -\-dy = 0. Divide by j/*. Then whence 3. Given (1 - a;*) yda-^il-y^ a?dy=0. Di- vide by a?y. Then \~a? , 1-1/ , at ■^ - dx -^ -{■ ydy = 0, INFINITESIMAL CALCULUS. 177 whence _l_a;-logy + | = (7, or 83. When the equation Mdx + ^dy = is homogeneous with regard to the variables, that is, when the sum of the exponents of the variables is the same in Jf as in iV, the variables cun be sepa- rated by the aid of an auxiliary variable. Let a?dy - y (x -}-y) dx = 0. • This equation being homogeneous, we assume yz=zx^ and therefore dy = zdx -\- xdz. Substitut- ing these values in the equation, we have a^zdx+ofdz — {foif + afz) dx = 0, and, dividing by ctf, xdz — fdx = 0, and -3- = — ^ :f X whence "1. X X log x=z k C= C ; and y = -j^ — ^ ® z^ y ^ (7— logrr In like manner, the equation %x^ I Xil x^y dy-ydx = being homogeneous, we assume y = zx, dy = zdx + xdz, and we find, by substitution and reduction, 178 INFINITESIMAL CALCULUS, x{l-^z) dz-\' 2z'dx = 0, whence — = — -~^ dz^ and ^^«^=i"*^^«^=£~^''^ 4/1+^- 2y Again, the equation ^ciy — yd^ = ^ ^^^ + y* being homogeneous, we assume y = zx, df/ := zdx -{- xdz, and we find, by substitution and reduction, * f79 d/X xdz = dx i^l-\- z% whence Accordingly loga; = log(2+f^l + 2') = log(|+yi + ^) + C, an integral which, freed from transcendentals and radicals, reduces to (7, being an arbitrary constant. 84. When Mdx + Ndy = is not an exact dif- ferential, it is possible to reduce it to an exact dif- ferential by an integrating factor , Thus (1 + 2/') ^^ + ^^2^ = becomes an exact differential if it be multiplied by ar, and gives the integral ^' (1 + 2/0 = c: INFINITESIMAL CALCULUS. 179 The multiplier 2a; is termed an integrating factor. The same equation might have been made an exact 1 differential by the integrating factor . ^ t » whence $ + 1^ = 0, and logrr + ilog(l + 2^) = 6V a result identical with the preceding one, though under a different form. The expression Mdx-^-Ndy can be written as follows : Since dx , dy J ^ ^ dx dy J ^ X f- -^ = ^ . log xy and ^ = a . log - , X y s.*/ X y ^ y we may write also Mdx-\- Ndy = ^ \{Mx + Ny) d:\og xyMMx^Ny)d.\og^Y (^) Nov/, if Mx-\- Ny happens to be identically = 0, we shall have Mdx + Ndy - , , x Mx — Ny * y and because the second member of this equation is an exact differential, the first member is also one ; that is, ^ ^ is an integiating factor. 180 INFINITESIMAL CALCULUS. If, on the contrary, Mx^Ny liappens to be identically =s 0, we shall have Mdx + Ndy , , , where, because the second member is an exact dif- ferential, the first member is also one ; that is, •Mf X j^ ' is then an integrating factor. But, if neither Mx-^Ny nor Mx— Nyi& iden- tically = 0, equation (a) divided by Mx+Ny gives Md'X'^-Ndy , , , ,1 Mx—Ny , , ^ /u\ -^^^ = ^d.\ogxy+,^j^^d.\og-, (b) and the last term of this equation will be an exact differential, if -ttp — :-i^ is a function of log - , or ' MX -\- Ny ® y X generally a function of - , that is, if it is a homo- if geneous function of x and y of the degree 0, as is ^ (-1. In such a case, then, the first member of (b) will also be a perfect differential, and the inte- grating factor will be ^^^j^r^ - Again, if neither Mx-^-Ny nor Mx — Ny is identically = 0, equation (a) divided by Mx — Ny gives MdxA-Ndy 1 Mx+Ny , , , i ^ , ^ / x INFINITESIMAL CALCULUS. 181 and the first tenn of the second member of this equation will be an exact differential if -jlt i^ is a function of log xy^ or generally of xy, that is, if it is a homogeneous function of x and y of the second degree, as is f {xy\ In such a case the first member of (c) will also be an exact differential, the integrating factor being j^^_j^y • To give an example of this process of integration, let (a?* +2/') ^^ "~ ^dy = 0. Here we have Jf=a^ + y*, and ir= — xy. Consequently Mx — Ny = af + 2x7/"^ Mx -^^ Ny •= :ji? ^ and Mx + Nv'^' x' ~ '^ \xj' + Ny Hence, by formula (b), {of + y^) dx — (cydy X' i^.loga:2/ + i(l+2|-!)^.log| = 0. y Integrating this expression, we have i log xy + k log I +y fi ^.log ^=C. X But, making - = ^, we find 182 INFINITESIMAL CALCULUS, ^ d.\oe - = / ^ ~ / z* ~ 22' ~ 2 U/ * And, therefore, the whole integral will reduce to As the proposed differential was homogeneous, we might have integrated it by assuming y = zx^ according to the method explained above (No. 83). We would thus have found dx = zxdz^ whence — = zdz, and X ' a result identical with the preceding one, as was to be expected. Integration of other Differential Eqvxitions, 86. We have seen in the preceding pages that our success in the integration of a differential fre- quently depends on our ability to give it a simpler form. Though we have pointed out many such cases, many others would have to be examined, if we had to give an adequate idea of the resources offered by the Calculus. But, as this is an elemen- tary work, we must content ourselves with giving a few examples of a certain number of other pro- cesses frequently adopted by analysts for the trans- formation of differentials not directly integrable. I. Let dy — aydx =^f {x) dx» (1) INFINITESIMAL CALCULUS, 183 If we make y=zuv^ we have dy = udv + vdu^ aud, by substitution, vdv + ^du — auvdx =/{x) dx. As one of the two quantities i^ and v can be arbi- trarily assumed, take vdu = auvdXj or du = at^iP. (2) Then the above equation will be reduced to udv —/ (x) dx. (3) Now, (2) gives log u = ax, or u = &^; and this, substituted in (3), gives , f(x) dx 6** Whence ''=/-^-+^. 3' =•"=«" U^^+o); which is the integral required. II. Let S-<.t + »!. = 0. (1) dx^ dx To simplify this expression, assume the two equa- tions ^^jcy^z, and ^-A:'2 = 0, (2) where z is an auxiliary variable, whilst k and Ar' are constants to be determined. Differentiating the first of equations (2), we have dx^ dx dx ' 184 lyPIMTESIMAL CALCULUS. or, since z=i-^ — Jcy^ substituting and reducing, which, compared with (1) gives A: + A:' = a, and kk' = b. Now, from the second of equations (2) we have — = k'dx, and z = 6»'*^^, or z= C^'%' z hence, substituting in the first of equations (2), and multiplying by Sir, ^y — kydx — Cte*'* dx^ an equation of the same form as the one which we have integrated in the preceding example. We shall have, therefore, But therefore (7, and C^ being two arbitrary constants. This same integral could be easily obtained by another method which deserves special notice, owing to its simplicity and the range of its applica- tion. In the equation INFINITESIMAL CALCULUS. 185 make y = C^<^. Then, differentiating, we have and therefore, substituting in (1), or, rejecting the factor, 6^6**, A:" — aA: + 6 = 0. This auxiliaTy equaiion gives k- 2 , a: - 2 , and these two values of Jc are connected by the re- lations A: + *' = ^» and kJc' = 6, as in the preceding solution. We have, therefore, two particular in- tegrals y = 0,61*'*, and y = C7,e**, which, added to- gether, will give us for the complete integral * If, in (1), we make a = 3, & = 2, we shall find * The complete integral of a differential equation of the second order mast not ooly satisfy that equation, but also contain two arbitrary constants (No. 62). The particular integrals y = (7|«*'«. and y = Ca«**, satisfy equation (1) ; for they give C7|*!*'« (*'«-«*' 4- 6) = 0, and » C,*»«(A«-a*4-&) = 0; but each contains only a single arbitrary constant. The integral which is the sum of these particular integrals, contains two arbitrary constantB, ind equally satisfies equation (1) ; for it gives ( C,«*'* + CV*«> (*« - a* 4- ft) = 0. Hence y =s Cie^'*-\-CfA^ is our complete integral. 186 INFINITESIMAL CALCULUS. A:' = 1,^=2; and the complete integral will be y = C,e" + C,e'. III. Let Assuming ?/=Cfe"**, differentiating, and substi- tuting in (1), we find Solving the auxiliary equation m' — 47?i + 13=0, we obtain ^' = 2 + 3^-1, m'' = 2-3i^-l; and the complete integral will be expressed by This integral, by referring to De Moivre's formulas (No. 26), may take the form Ce^ (cos 3ir + i^ - 1 sin 3x) + (7,6** (cos ar - 1/ - 1 sin ar) = ((7+ C,)^ cos 3x + {C-C,) i/ - 1 6^ sin 3x, or, replacing C-\-C, and ((7— C^) V —1 by new ar- bitrary constants, y = Aa** cos ar + ^^"^ sin 'Sx. This is the form of the integral, when the auxiliary equation has two unequal imaginary roots. IV. Let dhj d^y dy r7^'-^'-^+2/-0- (1) INFINITESIMAL CALCULUS. 187 Assuming p = 6V"% and differentiating, we find the auxiliary equation 7/^* ~ w* — m + 1 = 0, the roots of which are 7/^^ = — • 1, m, = 1, m^ = 1. Owing to the root w, = — 1, we shall have in the integral the term Cfe"*. The roots w, = 1, tw, = 1 will give the terms 0^6*+ ^»^*- But these two terms coalesce into the single term (C, + ^i) ^j where C,+ Ci is equivalent to a single arbitmry constant, whereas the complete integral of (1) must contain tliree arbitrary constants. To remedy the defi- ciency, let us begin by supposing tw, to differ from w, by a small quantity A. Then and, developing by the exponential formula (No. 24), C.er^'' + C,e^^ = er^ {g, + CJix+C, ^+ . . . ) Now, though h must become less than any assign- able quantity in order to verify the equation m, = w„ yet the product CJi may remain finite, if we assume (7, greater than any assignable quantity. Let, then, CJi = (7, be a new arbitrary constant. The complete integral will be and, in our case, with m, = — 1, m^-=-m^^=- 1, V. Let 188 INFINITESIMAL CALCULUS. Assume a? = ^, and y =ixz=.€fz^ d and z being two auxiliary variables. Differentiating the two as- sumed equations, we have dy = €fdZ'{'Z€^d^j dx=.efd&^ whence But a?" "" e^ - ^-^ '^^' therefore, substituting, and suppressing the com* mon factor e~^^ g+g-2. = 0. (8) To integrate this, assume z = Cfe"**. Then and (2) becomes Cfe'^(m' + m-2)=0. The auxiliary equation m* + ^ —2= gives m^ = 1, w, = — 2. Hence But ;? = - , and €^=zx. Therefore, finally, X l=Cx+^, or y=Caf + ^. HfFINITESIMAL CALCULUS. 189 VI. Let ^±«S+^2' = ^- ^1) To get rid of the term mx^ we shall assume y = z^Px+Q, (2) P and Q being constants to be suitably determined. Prom (2) we obtain dy _ ^ p cPy _ d'z dx" dx"^ ' dx* ~ dx* * ft These values substituted in (1) give ^±a'g±aP + bz + bPx-\-bQ = mx. (3) Now, let bP = m, and 6Q = T aP. Then (3) will be reduced to d*z , dz . , - ... ^±a^+6^=0. (4) and (2) will become 2/ = '2^+ J ^T-jr- (5) Equation (4) will be easily integrated by tlie method which we have followed in examples II and III. When z has been found, the integral of (1) will be known by equation (6). VII. Let 1 90 lyFINJTESJMA L CALCUL US. To get rid of the term e*, we shall assume y = e'{z + P), (2) P being a constant to be suitably determined. From (2) we obtain Substituting these values in (1), and cancelling the common factor c*, we have g-2(A:-l)g+(*-l)'^ + P(*-l)' = l. (3) Hence, if we make P(A:-l)« = l,orP = ^^^., we shall have g-3(*-»)g + (*-l)-.=0, and j,=^(,+ _i_). If we now make z = e^'', the integration of the last differential equation can be made to depend on the auxiliary equation m' -2m{k- 1) +(Ar - 1)' = 0, which gives 7n, = l' -^1= m,. Hence, by the rule of example IV, INFINITESIMAL CALCULUS. 191 and 3/ = c».c»-i)»(e+(7,aj) + <5» c*» (C+ C,x) + (*-!)• In this example, if A: = 1, equation (3) reduces to -^ = 1, and P disappears. In this case, equations (1) and (2) will become Hence, integrating the equation ^ = 1, we shall have for this case, z = '^ + Cx + C,, and y = e(^ + Cx+c). VIII. Let Assume y=iz-\ r — , ^ + n*y = cosax. (1) cos ax 1 , . J, J. X where A is a constant to be determined in view of a future simplification. We shall have dy dz a . 3^ = T T sm ax. dx dx a d'y d*z a" 3^ = T7i3 - r- cos ax. dx* dof h 192 INFINITESIMAL CALCULUS, Thus (1), by substitution, will become ^-; — J- COS ax-^-nz-^- ^- cos ax = cos ax, or -=-3 + n'i: = cos 00? 5 — cos aa?. dor ' Zt Take 'k-=:>7C — a', and the equation will reduce to d'z da? + n'z = 0. (2) To integrate this, assume z = c^, from which we find, by diflferentiation, wV"^ + Ti'e^ = 0, or w' + 71* = 0, and therefore, m^= ±nV -—l. Hence, by the method followed in example III, z = C^^~^ + C.e-***^* , that is, z=^A cos 7ix-\'B sin nXy and . . T> • • COS ax V=iA cos nx + B smnxA- -5 = , ^ ' ^ n —a A and 5 being arbitrary constants. Integration iy Elimination of Differentials. 86, There are differentials of which the integral can be found without direct integration. They are those whose form admits of a ready elimination of the differentials themselves. We shall show by a few examples how this method can be utilized. INFINITESIMAL CALCULUS, 193 I. Let Making ^=jp, whence dy^pdXy the equation becomes ^ ir = a + mp + np^. (2) Differentiating (2), we have &p = m^ + 8^i^'^> whence ^eftr, or dy = mpt^ + ^p^dp^ and yrziT/li^'+fTip^ + C'. (3) If we can eliminate p between (2) and (3), we shall have the expression of the complete integral. II. Let Making -^ =^, the equation becomes y = mp" + 271^'. (2) Differentiating (2), we have dy = 2mpdp + Qnp^dp ^pdx; whence dx = ^i/md/p + ^pd/p^ aiid therefore a;=:2^^ + 3wp' + a (3) Prom this equation we have - 77^± '^^nx + m' — 371(7 ^= 3^ 194 INFINITESIMAL CALCULUS. This value of p being placed in (2), we shall have the complete integral of the proposed equation. III. Let Dividing by 3^=i>j we obtain ^ 2/ = ^ + ~- (2) Differentiating, dividing by dx^ and reducing, we have {^-f)t='- ^ <^> This equation must be satisfied either by ^ = 0, or by a?-^ = 0. If ^ = 0, then p=C, and (2) be- comes 2/=C& + 5. (4) If X i = 0, then p=i Ju — ^ which value sub- stituted in (2) will give y^ = 4mx. (5) an integral without any arbitrary constant, that is, a singular solution ; * whereas equation (4) gives * By Hnguiar solutions we mean such resnlts as arc not contained in, or cannot be obtained from, the complete integrals, whatever particular valae be aeeigned to the arbitraiy constant. They can only be obtained by differentiating the complete in- tegrals with rsspectto the constant alone, coneidering it as a variable parameter, as we have done above (Nos. 49, 50), and substitnting its value, in terms of the vari- INFINITESIMAL CALCULUS. 195 the complete integral, and represents the tangents by whose intersections the parabola y^=zAmx is generated. IV. Let We write, as usual, rp-l-jpy — ap*=0 (2) whence dx -|- ydp +j>dy — 2apdp = 0. (3) From (2) we have y = — . Substituting this in (3) and reducing, we find dx (1 +^«) - a? ^ = apdp, which, being divided by pVl-\' p% will take the form xdp pdx Vl +^' - Vl+^' adp P VI +p' ' the integral of which is ablen, in the expression of the integrals. Thus, differentiating (4) with regard to C alone, .we find C= A/— i * value which will reduce (4) to y» = 4mx. It is not our purpose to discuss the relations existing between the complete in* tegrals and the singular solutions. We will simply state that, owing to such rela- tions, as interpreted by Geometry, the latter are often called envelopes of the former. This subject has been very deeply treated in the excellent work of Mr. George Boole on Differential Equatione. 1 96 INFINITESIMAL CALCULUS. or -^^^^' = a log (i? + i^l +i>') + C, ^ = 7ife5(^'+«^^8(^+^l+^'))- (4) Now, from (2) we have ^_ y± i^y' + 4aa? This value of ^, placed in (4), will give the com- plete integral. V. Let ydx — a>dy = n Vda? + dif. (1) Dividing by ete, and making ^=p^ as usual, we shall have y =:px + n vT+^, (2) whence dy=:pdX'\'Xdp-\--' ^^ Vl+p^ or, since dy=pdx^ ^(^+i7fei)=«- . (^) vT+p' This equation must be satisfied either by dp = 0, or by X + ^ =0. If eZ^ = 0, then p = (7/ and (2) becomes yz=Cx + nVl + C% INFINITESIMAL CALCULUS. 197 which is the complete integral. It xA — -^ — =0, ^ Vl-\-p' then ^ = ——-—. . which value placed in (2) VtC — of gives 2/- + ^ = < an integral without any arbitrary constant, repre- senting a singular solution. This sqlution is the locus of the intersections of all the straight lines obtained by varying the constant C in the complete integral. Dovhle Integrals. 87, When an infinitesimal area is referred to rectangular co-ordinates, its most general expres- sion is the infinitely small rectangle of which dx and dy are the sides ; and thus dA = dxdy. To find the area J., we must integrate between proper limits with respect to each variable in suc- cession. This double integration is indicated by writing the sign of integration twice before the quantity to be integrated. Thus we have the double integral A = // dydx. One of tliese integrations can always be per- formed, so that we may have either I ydx + 67, or j xdy + C. Then, if the equation of tlie line that bounds the area be y^=^f{x\ or x = we find w = A- Therefore tan-' -A = tan-» i + tan-' A- Assuming ^ = A, m = i, we find n--T^. There- fore tan-* A = tan-* i — tan-' ^. Hence, by successive substitutions, we at last find tan"^ = - = 4 tan"^ \ — tan"^ ^^. Thus, if we make x^\\n the formula, we find the value of tan-^, and in like manner, if we make 0? = ^T we find the value of tan'^ -^^ The calcula- tion of six terms for tan-*|, and of two terms for tan"^ irfr? will give the value of ^ np to the tenth decimal. 91. The differential of an arc, when expressed in function of polar co-ordinates, is (No. 52) # ds ~ Vcip"" + p'dip\ p being the radius vector, and (p the angle which p makes with the initial line. The integration will be made as follows. INFINITESIMAL CALCULUS, 203 Example I. To find an arc of the spiral of Archimedes. The equation of the curve is /> = ay, where a — ^r- Hence d(p = — . and Integrating by i)arts, and the last term, integrated by formula (33), gives Hence ^ = h,P ^"^P^ +1 log 0* + V7+T') + C; and taking Hie integral from p = to p = r, we have for the first spire or, since a = ^r- , ^ = 1 1/1 + 4? + 1^ log (2;r + vT+i?). To obtain the length of n spires, it suffices to take the integral from /t> = to /> = nr. Example II. To find the length of the curve traced by the end ^ of a tense string AB (Fig. 33), whose other end A is fixed on a circle around which the string is being wrapped. i 1 Ftflf.W 204 INFINITESIMAL CALCULUS, Let AB = ?, and AC-^r^ and let ^ be a point on the curve. Draw ED tangent to the circle, and draw the radius CD. Make ACD^^, and b consequently, the arc AD = r^. Then the in- finitesimal arc ds = EF described by ED will be equal to ED X d&. But, by the nature of the case, ED ^AB — AD ^l-^r^. Therefore Integrating from i> = to i> = i?, we have As a particular case, assume I = 27tr ; then the whole curve will be described when * = 2n. Then *=2;r'r,or^ = 2j.. Quadraiure of Curves. 92. The differential of the area of a curve re- ferred to rectangular axes is (No. 51) dA = ydx. This equation, integrated between proper limits, will give us the area intercepted between the curve, the axis of rr, and the limiting ordinates. Example I. Let the curve be a x>arabola y' = 2px. Then dx = ^-^ ; whence P dA = t^ P INFINITESIMAL CALCULUS, 205 Integrating from y = Otoy=^yy we have Example II. Let the curve be a cycloid. Then dx==— —=====. whence v2ry — y* rfl ^^^ V 2ry — y* Integrating by formula (37), we have f V'dy ^ _ p V2ry - y' , 3r / " y% . y \f2ry-y^ 2 "'' 2y V^ry — y' and again, by tlie same formula, f—^^- = - 1/27-2/ - «/• + r vers-' ^ • I Substituting this value in the last term of the pre- ceding equation, and taking the integral from y = to y = 2r, we have -A = TT- vers"^ — = -^r- • 2 r 2 This is the area of a semi- cycloid. The area of the whole cycloid is therefore 3;rr', or three times the area of the generating circle. Example III. Let the curve be a circle. In this case, since y = Vr^ — af , we have dA = dx Vr' — a;' whence (No. 73) 206 INFINITESIMAL CALCULUS. Taking this integral from a; = to a; = r, we shall have A = \nr\ for a quadrant, and therefore 4-4 = ;rr' for the whole circle. Example IV. Let the curve be the logarithmic y = log X. Then dA = log xdx. Integmting by parts, we have / log xdx = X log X— / dx = x (log a: — 1) -+- (7. Taking this integral from ^ = 1 to a: =x , we have A =x log a? — re + 1^ for the area extending above the axis of re, so long as we take x>l. But if we take the integral from a; = to re = 1, then we find ^ = l~[rr(loga:-l)]o = l; for we know (No. 37) that x log a? - a? = when x=0. Thus A = lis the area extending to infin- ity beneath the axis of x from a? = 1 to a? = 0, that is, from y=0tO2^=— c». Example V. Let the curve be the equilateral hyperbola xy = m". Then y = — ; hence X a A =zm — , X and integluting from a? = 1 to a? = a?, A=m*logx, INFINITESIMAL CALCULUS, 207 When m = 1, then A = log x. Thus the Napierian logarithms, whose modulus is = 1, exhibit so many areas taken in the hyperbola xy = l. Hence they are also called hyperbolic logarithms. 93. When the curve is referred to polar coordi- nates, then (No 52) dA =z ^p'dip. In the spiral of Archimedes^ in which p = af , and dp = a^f , we have dA = \(^ip^dAp^ or dA = ^ p^dp. Integrating the first expression from ^ = to ^ = 2;r, and the second from* ^ = to /> = r, we have ^ = I' (2;ry, and ^ = |.^'; r and, as a = ^ , both expressions reduce to -4 = J;rr'. In the parabolic spiral^ in which />' = 2af , and pdp = adipj we have dA =a(pdf^ and integrating from y = to ^ = 2;r, A = 2a7r\ r* But, as a = J- (No. 56), this surface becomes 47r 208 INFINITESIMAL CALCULUS, In the hyperbolic spiral^ in which pip = a, and di(>= r » we have dA = ^^p' -Y = -^^P' And integrating from p=pto p^a^ a being the radius of the measuring circle. In the logarithmic spiral^ in which f = a log />, and d

= 1 to p = />, Surfaces of Hevolution. 94. The area of a surface of revolution will be found by integrating the differential expression (No. 51) dS = 2np Vdx* + dy". Example I. Shirface of a cone. The convex sur- face of a cone is generated by a straight line y = ax revolving about the axis of x. We have, in this case, dy = adx; hence INFINITESIMAL CALCULUS. 209 dS = indxdx f'a' + l ; and integrating from x = Oiox:=h^ 8= ncih^ Va' + 1. T Let r be the radius of the base ; then « = t ; and 5f = 2;rr . —■ — ; which is the common expression of iS^ in Geometry. Example II. Surface of a sphere. From the equation y=iV2rX'-af of the circle, we have dy = '- (r — x)dx: hence ^ y dS^27ty J da? + ^^ f^^ da? = ^Ttrdx; and integrating from x = to x = 2r^ 8=47rr\ which is the surface of the sphere. Example III. Surface of a paraboloid of revolu- tion. From the equation y* = 2px of the parabola, we have dy = - — ; hence if dS= 2n V2px a/(i J^Q daf = 2ndx V2px+p%' and integrating from a? = to .r = ^r, 210 INFINITESIMAL CALCULUS. Example IV. Surface generated by a cycloid revoloing about its base. In the cycloid we have dx=.^jJ^=; i^2ry — y" hence, by substitution and reduction, ydy dS =27: i^ V2r — y or, making 2r — y = -?•, whence y=2r — f^ and dy= — 2zdZy dS =-47r V2r (2r - z').dz, $ind 8=-47r^2r\2rz-^J, or 8 = — 47tV2r (2r V2r-^y - J i/(2r-2/y) + C'. Taking the integral from y = to y = 2r, and re- ducing, we get S=^7rr\ Hence the whole surface generated will be = A^ ;rr\ Solids of Resolution. 96. The volume of a solid of revolution will be found by integrating between proper limits the differential (No. 61) d F= Tzy^dx. Example I. Volume of the frustum of a cone. The cone is generated by the revolution of a line INFINITESIMAL CALCULUS, 211 y^ax about the axis of x. Hence we have jy» = aV. Substituting this value of y', and inte- grating from a? = 7^' to 0? = A, we find for the volume of the frustum y= It % {JC - h") = Tta' ^^' (/*♦ + h?i' + A"), where 7i — 7i' is the altitude of the frustum, Trd'h* its lower base, ;ra'A" its upper base, and na^hh' a mean proportional between the two bases, accord- ing to a well-known theorem of Geometry. Mak- ing A' = 0, we lind for the volume of the whole cone, K = T:y -^ . Example II. Volume of the sphere. Tiie equa- tion of the generating circle being y" ^r^ ^ a?^ our differential formula becomes whence, integrating from a; = -- r to re = r, T7 /o . 2r«\ 4nT« Example III. VoluToe of the prolate spheroid^ which is generated by the revolution of an ellipse about its transverse axis. We have y' = -i {a* —of)] hence and integrating from a? = — a to a: = a. 212 ISFIMTESIMAL CALCULUS. Example IV. VoluToe of the oblate spheroid^ which is generated by the revolution of an ellipse about its conjugate axis. In this case, the differen- tial of the volume is d T = na^dy. Now hence and integrating from y z=z ^h to y = 5, y = 3 Example V. Volume of the paraboloid of reoolu- turn. The equation of the revolving parabola being y* = 2pXy we shall have dV=27rpxdXj and integrating from a: = to x = Xj X V=7rpaf=z7ry'.^, which is equal to the volume of a cylinder having the same base, and half the height of the parabo- loid. Example VI. Volume generated hy the revohi- tion of a cycloid dbotU its base. Since, in this case, dx ydy V2ry — ^ the differential becomes INFINITESIMAL CALCULUS^ 213 V2ry — y" By formula (37) applied three times in succession, we shall find V= Taking the integral from y = to 2/ = 2r, we have for one-half of the volume hence the whole volume will be = 5;rr' X Jrr. O^A^ Oeometrical Problems. 96. Problem I. To find the curve whose svb- tangent is constant. The expression for the subtangent is (No. 34) ^— . Hence if a be the constant, dy Consequently, — = — , and x = a log y-f C. This equation shows that the curve is a logarithmic. Problem II. To find the curve whose svhnormal is constant. The expression for the subnormal is (No. 35) ^-^ . \ 214 lypiyiTESiMAL calculus. Hence if a be the constant, dx Consequently, ydy = adXy and y" = 2ax + C. The curve is therefore a parabola whose parame- ter is 2a, and whose vertex lies anywhere on the axis of X. Problem III. To find the curve whose nor Trial is constant The expression for the normal is Hence, if a be the constant, y 4/1 + :r^ = a, whence (fo? = ± ^ ^ , and integrating, or (a:-67)' + y' = a«. The curve is therefore a circle having the radius a. Problem IV. To find the curve whose tangent is constant. The expression for the tangent is ds A /i X ^^' INFINITESIMAL CALCULUS. 215 Hence if a be the constant, y y 1 + ^ = a, whence dx^±-^ Va^ - j/*. Multiplying and dividing by Va^ — y", we find and dy X = ±Va«-2/'±a/^;-;^== zdz Make a* - 2/* = -s:*? whence e?y = ; ; then / dy _^ f ^^ _ y Va' - 2/' "" y d^-^~ 2ay Va + ^ ' a— 2/ 2a ^ a — z But - log —— ' = - log — ■ ^ . And therefore «. /- __^_= _ a log i±-*^HZ Consequently 216 INFINITESIMAL CALCULUS. The curve of this equation is called the tractrix. Problem V. To find the curve in which the square qf the arc is proportional to the ordinate. The equation of condition will evidently be s* = ayy a being a constant. Differentiating, we have 28ds = ady^ , ady ady , whence d^ = dy ^ ^--\. Ay Making a = 8?*, we shall have dx =. dy i/i— #y ?2^ = |,/WF^, whence dy _ y dx \^2ry — ^ Now, this equation differs from the differential equation of the cycloid in this only, that we have d^ dx -5^ instead of 3— . It follows that our equation dx dy ^ belongs to a cycloid whose vertex has the axis of x for tangent. Hence the curve is a cycloid gene- rated by a circle whose radius is r = ^ . INFINITESIMAL CAWVLUS, 217 Pkoblem VI. To find the curve whose evolute is a circle. Let ^5=r (Fig. 34) be tlie radius of a circle, and let the curve start from the point JS. Let M be a point in the curve, whose co-ordinates are a/ and y\ Draw MT tan- gent to the circle, and therefore normal to the curve ; then MT will be the radius of curvature at the point Jf, as is plain from the theory of evolutes (No. 47). Now, the equation of the normal is (No. 35) and since the line passes through the point jT, whose co-ordinates we shall designate by a and ^, assuming rp = a and y = ^, we shall have (1) But, as MT is tangent to the circle at the point («i /5)5 we have also from the general equation of the tangent (No. 34) ^-2/'=f («-n Therefore da dx' _ d^ dy''^ da 218 INFINITESIMAL CALCULUS, Now, since a" + ^ = r", therefore a^ + /3t?/3 ==0, :, d3 a ^ ^, dx' a • . and -5r- = — ^ ; and consequently -j-, — ^ , or, omit- ting the accents, and clearing of fractions, ady — fidx = 0. (2) On the other hand, equation (1) will become and this, owing to the relation a" + i^" = r', reduces to or + ^2/ = r\ (3) Prom (2) and (3) we obtain — ^*^ R — ^'<^y " a>dX'\-ydy^ ^~ xdx-^ydy' Squaring these equations, adding them together, and changing «" +;? into r\ we find {xdx + ydy^ = r" {da? + dy"). Let p be the radius vector AM. Since jo' = af-\-y^^ we shall have pdp = (tdx -{- ydy ; also, if BM=s, we shall have ^fa?" + ^2/* = ^^'- Therefore, substi- tuting, and extracting the square root, pdp = rds (4) whence p' = 2rs+C. When ^ = 0, then p=r. Therefore C=r^; and accordingly p*is:2rs-\- r*. INFINITESIMAL CALCVLUS, 219 Let the angle MAB = (p ; then ds = Vdf^* -\- p^dip^ ; and (4) becomes pdp = r Vdp*-{' p^dif^y whence rdif = -^ V/>' — r" ; and multiplying and dividing this by V/o* — r*, and integrating, When f =: 0, we have p^r\ hence (7 = — ^ ; and therefore ^ = - i^^» — r' — cos'^ - . (5) Such is the polar equation of the curve required. To find a corresponding equation with rectangu- lar co-ordinates, we may remark that equation (5) gives T } or ^ = cos (V/>«-r"-f), 1 .-= — ^ . . . 1 - = cos f COS - '^p^ — r' + sin if sin - ^/p^ — r". But P^ = x'-\-y\ cosf = ^, siuf = ^. r r tSFINITESIMAL CALCULUS. Therefore r=jr cos (- Va^ + y'-r'^-^-ysm v'a^+y'-r') ■ Saoh is the equation of the curve with rectangular co-ordinates. The radical Va^ + y* -r*= TM re- presents the radius of curvature. Fbobleh VII. A ship starting from a given lati- tude erfect sphere and that the ship finds no obstacle in its way. LetOP=7-(Fig. 35) be ■the radius of the globe. When the ship reaches a place A in latitude X and longitude i?, let A C be the direction of its course. Taking A C in- tinitesimal, we shall have AB — AO'.d& = r cos X . d&, Ft(i.SS BC = BO.dl=rdX; JC' = ds^ = r' cos' Mff* + T'dX\ (1) Since CAD= a, we have AB = BC tan a, that is, INFINITESIMAL CALCULUS. 221 r cos Xdd^ = rdX . tan a. (2) This being squared, and substituted in (1), we have r'dX' hence efo' = rWA'(l + tan'a)= , , ^ ' ^ cos a ' dS=: , S = .X+C. COS a ' COS a As 5 = when >l = ^, we shall have COS a ' and therefore cos a /7i Prom (2) we obtain d^ = tan a z ; whence cos A (No, 77) ^ = — tan a log. tan ^ (o ~ ^) + ^• But, when # = ??o? then A = /^ ; hence C= i?o + tan a log tan 5 (2 "" ^) ' and therefore tan 2 (I - A,) * - 1?, = tan a log J— ^ ; (4) **^ 2 (2 - ^) from which we obtain 222 lyFINITESIMAL CALCULUS. A=5-2tan-» ^""l^^-^) >(«-lb>0Ot« and ;-;io = (^-;io)— 2 tan -1 tan 2 \2 V >(f-#o)eotc (6) This value substituted in (3) gives and the problem is solved. As a particular case, assume Then the ship starts from the equator in the direc- tion north-east or north-west, and traverses all the meridians. Then we have * = log tan la-)' n Make 2 tan~^ ^^^' Then e^ = cot J« ; and tak- ing the logarithms, log.cot i^ = 2;r.log e = 6,283186 X 0.434294 = 2.728740, INFINITESIMAL CALCULUS. 223 which corresponds to the cotangent of 0°6'26".2. Therefore z = (f 12' 50".4, and ;i = 90** - 0° 12'50^4. The ship will therefore be in latitude 89^ 47' 9^6. Calculating the value of s and assuming the radius of the earth = 3,960 miles, we shall find ^ = 8,774 miles. The assumption ^ = ^ gives i? = oo ; hence the ship can never reach the pole. Indeed, if its course constantly makes an angle a with the meridian, it is plain that whilst the meridian traverses the pole, the ship must always be at either side of it. Problems solved by Doiible or Triple Integrals. 97. A few examples will show how geometrical problems may be solved by double or triple Inte- gra don. Problem I. A circle AMB (Fig. 36) revolves about the axis OZ. To FUj-'Se ^ find the surface gene- rated by the circumfer- ence of that circfe. Let CM=^r, CN=x, ^ NM=y, and OC^R. The distance of the point -Sf from the axis OZ will be-R+a;/ hence the in- ^ finitesimal element ds of the circumference, while describing an infinitesimal angle d&j will generate a surface dS={Ii+x)d»ds. 224 INFINITESIMAL CALCULUS. Therefore 8=/yiji + x)ddds. We integrate first with respect to ^, which is in- dependent both of 8 and of x. The integral from i> = 0tot>=2?rwillbe 3=27: f{Ji + x)ds; or« since /dx that is, 8 = 27rr ^Ji sin^^ ^^ i^?^^^^) + C. And taking the integral between x=z — r and xz=z + r, S = 27rr {H sin-^ 1 - ^ sin** - 1) = 2zr . ;r JS. This is the area of the surface generated by the semi-circumference AMB, The surface of the whole ring is therefore =2;rr.2;r5. Problem II. The axes of two equal right cir- cular cylinders intersect at right angles. To find the volume of the solid common to both. Let us take the origin of co-ordinates at the in- tersection of the axes, the one being the axis of x^ the other of y ; and let r be the radius of the base of the cylinders. The equations of their surfaces will be INFINITESIMAL CALCULUS. 225 whilst the volume of their common part will be ex- pressed by V = fffcixdydz. Integrating first with respect to x^ and reflecting that the integral must begin from a; = 0, we have V = ffxdydZj or, substituting for x its value Vr" — z"^ V=fjdydz Vt^ - z\ Integrating now with respect to y from y = to y = Vr^ — jgr*, we have V=J^dz{r'-s^). Integrating finally with regard to ^, from ;2; = to 2 = r, we have This is one-eighth of the intercepted solid. The whole IS = -K— • Problem III. To find the volume qf the ellip- soid with unequal axes, whose equation is 226 lypiyiTESiMAL calculus. or The general formula being integrated from -2 = to z^-z^ then substi- tuting for z its value, we have Integrating with respect to y, we have — i — J i — «• .j_|^5i(^_^ 6' («' - g*) . 1 y a And, taking the integral from y = to ^ = S^' and reducing, V= I . ^y(a' - of) dx. INFINITESIMAL CALCULUS. 227 This, integrated from x = Oto x:=a^ will give T^ _ ;r he 2a* _ n abc ^ '~l'a^"3'~2' "3" ' and, as this volume is only one-eighth of the wliole ellipsoid, the entire volume will be 4tJTabc If a = 6 = c, the ellipsoid becomes a sphere, and ^^ 47ra* 228 INFINITESIMAL CALCULUS. SECTION III APPLICATION OF INTEGRAL CALCULUS TO MECHANICS. 98. Before attempting the solution of any me- chanical problem, it is necessary to give an exact definition of Force. This word Force is often pro- miscuously used as synonymous .with quantity of action^ quantity of movement^ quantity of pres- sure^ quantity of deceleration^ etc., and a great deal of confusion has been created by this loose employment of the word. When an agent exerts its power to produce or modify movement, it acts ; and the quantity of its action is a dynamical force^ which is measured by the change that would be produced in the quantity of movement if the action lasted for a second of time. A quantity of movement is a kinematic or kinetic force ; it is measured by the product of the actual velocity of the movement into the mass in motion. A quantity of pressure is a statical force ; it is measured by the quantity of movement that the pressing mass would acquire in the unit of time if all obstacle to motion were suppressed. Though a body in movement can do work^ as we shall explain, yet movement, as such, is not a INFINITESIMAL CALCULUS. 229 IfcM^^I^— Wi^Wi^i^^^^'^— IMII ■ ■ Will !■ I W IM^^W^— ■» !■ ■■■■■■■ Mil II I »l I ■■■ I p ■ ^ I ■■! ■!■■■ I ■ ■ 1 I I ■ ■ I i^W^— ^— force, but only a change of position due to an exer- tion of power. Velocity is the act or form of movement ; hence, when a body moves with a con- stant velocity, its movement is said to be uniform. Velocity has intensity^ but the space measured by the body has only extension. Time is the actual- ity, or duration, of movement ; and if the movement be uniform, time is the ratio of its extension to its intensity, that is, the ratio of the space measured to the velocity with which it is measured. Velocity, on the other hand, is measured and ex- pressed by the length which it causes to be meas- ured in the unit of time ; in other words, the intensity of the movement is measured by its ex- tension in the unit of time. Hence, if the velocity V remains unaltered, the space s measured in a number t of seconds, will be 5 = vt. Velocity is always gradually acquired, or grad- ually lost, through a series of infinitesimal incre- ments or decrements corresponding to the series of infinitesimal instants during which the body is acted upon. This is true even in the case of the so-called instantaneons forces^ v. gr., in the com- munication of velocity by impact. Even in this case, the action is really continuous ; and the only reason why it is called instantaneous is, that its continuation is too short to allow us to value ex- actly its duration. This short duration we often call an instant^ though it is a finite length of time, and comprises a series of infinitesimal instants. It follows, that all velocity is gradually acquired, or lost, by infinitesimal degrees, through some con- tinuous exertion of power. It also follows, that a constant continuous action, all other things being 230 INFINITESIMAL CALCULUS. equal, is proportional to the length of time during which it is allowed to continue. If, then, an agent by its continuous action is competent to impart to a body a velocity v in the unit of time, the same agent, all other things remaining equal, will in the instant d^ impart a velocity vdt; for l\ dtv.v : vdi. Let us conceive a fi'ee material point, which under the continuous action of an agent A meas- ures a space rr in a time ^, and at the end of this time has a velocity v. If left to itself, the point will, with this velocity, measure in the following infinitesimal instant dt an infinitesimal space dx; and as movement during an infinitesimal instant cannot but be uniform, we must have ' dt But, if the agent A continues to act after the time ty then the velocity v in the instant dt will undergo an infinitesimal change dv. Let, then, a represent the intensity of the action at the end of the time <, Since this action will in the instant dt produce the change dv, we shall have adt = dVy or a : dv::l : dt; which means that the actions, all other things being equal, are proportional to their duration. From adt = dv, we have dv d^x a = ^, or a = ^, which is the general expression of the action that modifies the velocity of the movement. Its effort d^x ■js- is called acceleration. d^ INFINITESIMAL CALCULUS. 231 If, instead of one point, we have a mass contain- ing a number ilf of points, and if the agent A acts equally on each of them, its total action will be and this is called a dynamical force^ or better, the quantity of the action after tlie time t It is the product of the acceleration into the mass of the body acted on. The quantity of movement of the mass M at the end of the time t^ is evidently Mv^ or M-^ ; and this is a Mnetic force. It is the product of the mass into its velocity. The quantity of pressure, or the statical for cCy is expressed by Mv like the kinetic force, v represent- ing the velocity which the mass M would acquire in a second of time, if no obstacle existed. It is the product qf the mass into its virtual velocity. WorJc. 99. A body moving under a continuous resist- ance is said to do worTc. The work is by so much the greater according as a greater mass measures a greater space under a greater resistance. Hence the unit of work will be the work done by the unit of mass, measuring the unit of space, under a unit of resistance ; and, if m be any mass, li the resist- ance, 8 the space measured, W the work, we shall have W=m.Ii.S. 232 INFINITESIMAL CALCULUS, We are going to show that, if u be the initial velocity of the mass m, the total work which this mass can do, is always = ^u^. When the moving mass, after a time t^ has measured a space x^ its velocity may be expressed by -jj ; and the resistance, which is an action modi- fying that velocity, and opposed to it, will be ex- pressed by — -j^ . Hence the infinitesimal work dW corresponding to the instant dty will be the product of the three factors, w, dx^ and — -^ • Therefore dW= — mdx -js- • d^ Integrating this, we have dil2/ But, when < = 0, we have TF = 0, and -^r =^ u. Therefore 0= ^u*. Hence W= imu^ — im (-^J • When the total work has been done, then dx 3/ = 0. Therefore the total work is W = imu". The total work of which a mass is capable is often designated by the name of energy^ or work stored up in the mass m. INFINITESIMAL CALCULUS. 233 Example I. Work under constant resistance.' A body m is thrown up vertically with an initial velocity u. The resistance is here the action of gravity, which we represent by the letter g. Hence we have Multiplying both members of this equation by dx^ and integrating, ^=-Ht)"+« When t=0, we have a; = 0, and -^ — u. There- fore C = i^'; and /To* But, when the whole work is done, then ^ = ^) and ic=51 Therefore g8^\u\ or, multiplying by w, Example II. Work under a resistance varying in the inverse ratio of the squared distances. Let a mass m with the initial velocity u recede from a centre of attraction whose action at the unit of dis- tance we shall designate by A. Let a be the original distance of m from the centre of attrac- tion. Then, every particle of the mass m, after a time t^ will be subject to a resistance or retardation 234 INFINITESIMAL CALCULUS, 1? _ ^'^ -. Hence dx d?x dt di^ (a + a?)" 2 V^/ "a + a:"'"^* When rr = 0, we have -rr = w / hence (7= ^ dt ' a Substituting, and reducing, Ax 1 /^Y - i • ^ 2 WJ ~*^ a(a + ^) When the whole work has been done, then ^^ = 0, ' dt ' and x = S. Hence . ,. _ AS . and multiplying by w, a (a + iS) * ' where the resistance —7 — r— ttv is a geometrical mean a{a + S) ° between the initial resistance —, and the final re- sistance (a + sy Example III. Resistance proportional to the velocity of the movement. In this case, if a be a constant, we shall have d'x dx d^x __ , INFINITESIMAL CALCULUS. 235 dt ' dx When a? = 0, then -^^u; hence C=:u; and therefore dx -^T^u — ax. dt ' whence dx 1 dt = , ^ = log (u — aa?) + C\ u — dx^ a ^^ ' ' As <=0 gives ic = 0, hence C^' = - log t^. There- fore , 1 , u < = - log a " t^ — Oic / d/X When the whole work has been done, then ^ = 0, x=S=^ - , and t= co. It would, therefore, take a ' infinite time to measure the finite space S^ and to exhaust the velocity u. u Introducing the value S=- into the general equation we find H = iau. Thus the mean resistance is here an arithmetical mean between the initial resistance au and the final resistance zero. Therefore the mass m would, under a constant resistance ^au measure the space 8= - > 236 INFIMTESIMAL CALCULUS. Example IV. Resistance proportional to the spa^e measured. In this case, if a be a constant, we liave cTx dx (Px J (t)"=-«' + « cLx When a: = 0, then -=j =t^. Hence C=^u^. There- ' dt fore (f )■ =«•--* dx When the whole work has been done, then ^ = 0, X = — = = 8. Hence Va m,8,Ii=m -% Ii = imu*: Va and therefore R^^i^ui^. This mean resistance is an arithmetical mean between the initial resistance zero and the final resistance u Va, It may be observed that the integration of <=-7= sin-* - Va^ Va ^ ' which, when x=:S=z -— , reduces to va INFINITESIMAL CALCULUS. 237 -_sm"*l This shows that the time employed in doing the whole work would be independent of the initial velocity, and of the space measured. Movement Uniformly Varied. 100. When the velocity of a mass m increases or decreases uniformly, its movement is uniformly accelerated or retarded ; which implies tliat the action, under which the movement varies, must be constant. Assume m = l, and let the accelerating action be g. Then the equation of the movement will be ^ = g' / whence s being the space measured at the end of the time ds t The constant O is what the velocity -=^ becomes when t = 0; that is, is the initial velocity. Hence, if there be no initial velocity, C will be = 0. The constant C is what s becomes when i? = ; that is, C" is the initial space^ or a space already measured before the beginning of the time t Hence, when the space is reckoned from the begin- ning of the time t, C will be =0. Supposing therefore ^^=0 and (7' = in the above equations, the velocity v acquired in the time t will be v = 9t (1) 238 INFINITESIMAL CALCULUS. and the space traversed in the same time s = \gt\ (2) This last equation, being multiplied by 3p^, gives 2g8 = ^r, or ©• = 2gs, and v = V2gs, (3) that is, the velocity acquired or lost by a body while measuring a space s with uniformly accele- i*ated or retarded movement, is equal to the square root of twice tlie product of such space into the accelerating or retarding action. Such a velocity is styled the velocity dv£ to the space s, whilst s itself is called the space dit£ to the velocity v. Movement not Uniformty Varied. 101. The attraction of the earth, and of planets in general, varies inversely as the squared distances of the bodies attracted. Let r be the radius of the earth, g the intensity of its attmction at its surface, g' the intensity of its attraction at a distance s. Then g' \ g \\ T^ : ^*, whence g'=^g -i • s We shall have, therefore, where the second member has the negative sign, because the action tends to diminish the dis- tance s. INFINITESIMAL CALCULUS, 239 Multiplying this by 2d^, and integrating, we have (iy=^^^« ds Let s=ih when t=Q ; then ^ = 0, and hence Such is the velocity acquired by the body while falling from the original height Ji to the height s. Let A = 00 , and « = r. Then equation (1) be- comes t* = 2gT, hence v = V^r, and, as ^r = 32.088 feet, r = 20,923,596 feet, we shall find 7) = 36,644 feet. Thus a body falling upon the earth from an in- finite distance would reach its surface with a veloc- ity of nearly 7 miles a second. The attraction of the sun at its surface is g = 890.16 feet, and its radius is r = 430,854.5 miles. With these data, we find that a body falling upon the sun from an infinite distance would have a final velocity of about 381 miles per second. From equation (1), extracting the root, and tak- ing the negative signs (as ds diminishes w^hen dt sds Vhs -s' ' 240 INFINITESIMAL CALCULUS. increases), we obtain which may be written thus,* h h — ( ^ds h J 1 {7i - 2s) ds _ 2 ^9r'i2' i/hs - s* Vhs^^' Hence V: ~Vw \ Vhs - s*- 1 vers-* j\-hO. But, when t=0, we have g = h. Therefore k ./ h = 2'' y25^ 2gr* And therefore This equation, when ^ = r, will give the time re- quired for a body to fall from a distance Ti to the surface of the attracting body. The mean distance of the moon from the earth is 60r, r being the radius of the earth. The time re- quired for a body to fall from the moon to the earth would be ^ = 417, 381 seconds, that is to say, 4 days, 19 hours, 66 minutes, and 21 seconds. INFINITESIMAL CALCULUS. 241 Composition and Decomposition of Forces. 102. Before we proceed further, we must say a few words on the composition and decomposition of forces. Dynamical forces are usually repre- sented by lines proportional to the velocities which they impart in the unit of time ; kinetic forces by lines proportional to the spaces measured in the unit of time ; statical forces by lines proportional to the spaces that would be measured in the unit of time, if no obstacle existed. Let two equal forces P and P (Fig. 37) be applied to a point A^ and let them make with each other an angle PAP = 2x. If we can find a single force -ff, which applied to A will produce the same effect as the two forces P, this new force will be called the re- sultant of the forces P, whilst the forces P will be its components. The resultant will evidently lie in the plane of its comprnents ; and when the two components are equal, it will bisect their angle. The value of the resultant depends both on the intensity of its components, and on the angle at which they meet. Hence the resultant R of the two equal forces P may be expressed by R = iPf{x\ f{x) being a trigonometric function of the angle x. This function is easily determined. For we know, that when x = 0°, the resultant is ^ = 2P. Hence /(0°) = 1. We know, also, that when a;= 60°, the 242 INFINITESIMAL CALCULUS, resultant is R = P. Hence/ (60°) = i. We know in like manner, that when x = 90°, the resultant is ^ ~ 0, as the two forces then neutralize each other. Hence /(90°) = 0. Now, the only trigonometric function which can satisfy these conditions is the cosine of x; for cosO° = l, co8 60° = i, cos 90° = 0. Therefore the resultant of the two equal forces P is R = 2P cos X. Drawing PP. which will intersect AJi at right angles in Z), we have AD = P cos x; and there- fore i? = 2^Z) — AJi, Thus the resultant is repre- sented as to its intensity .and direction by the diagonal of the rhombus constructed on the two equal components. Fig.SS 103. Let now two unequal forces P and Q be applied to the point J. at right angles (Fig. 38), and let X and y be the angles which tlie resultant shall make with P and Q respec- tively. If P be conceived as a resultant of two equal forces p that make with it an angle a?, one of these components will lie in the direction of the resultant, and the other in the direction Ap. K, in like manner, Q be conceived as a resultant of two equal forces q that make with it an angle y, one of these components will lie in the direction of INFINITESIMAL CALCULUS. 243 the resultant, and the other in the direction Aq. And we shall have (No. 102) Pzm'ip cos Xy Q = 2q cos y. Now, evidently, the resultant is the snm of the two forces p and q which lie in its direction, while the other two forces p and q which lie in the direction Ap and Aq^ are directly opposite, and must neu- tralize each other. Therefore p =q, and H =2p=2q. Hence i2= ^ - « • cos X COS y or R cos x=P^ H cos y = Q. Squaring these equations, adding them together and remarking that in our case cos y = sin x, we obtain R'=P' + Q', that is, the resultant of two forces meeting at a right angle is the diagonal of the rectangle con- structed on its components. 104. If the forces P and Q do not form a right angle (Fig. 39), then considering P as the diagonal of a rectangle of which ^^ one side Ap lies on the , ' resultant, and Q as the ^ ~^Z^^^^^^-^ diagonal of a rectangle \^y^ j of which one side Aq Ajhus we may write in general When the point lies within the angle formed by the resultant and any of its components, then the moments of such components are negative ; for sin {ip — a) is then negative. Hence the above sum of moments is algebraical, not arithmetical. Fig.4S Virtual Moments. 107, When two forces P and Q applied at the extremities A and ^ of the lever AB (Fig. 43) are in equilibrium about the A/ fixed point 0, if the equi- librium be disturbed or endangered by an ex traneous force, the points A and B will describe, or tend to describe, similar arcs A A' and BB' about the point ; and we shall have the proportion AA' :BB'::OA\ OB. Projecting AA' on the direction of P, and BB' on the direction of Q^ we shall have also Aa:Bb::AA' :BB'\ 250 INFINITESIMAL CALCULUS. and therefore Aa: Bb::OA : OB; and because OA : OB::Q : P, tlierefore Aa: Bb::Q: P, or P.Aa = Q.Bb. The lines Aa and Bb represent the so-called virtual velocities of the forces P and Q. Not that these forces have any velocity at all, but because Aa and Bb are the measure of the virtual velocities which rule the ascent or descent of the points of application of the forces. The former, Aa^ which falls on AP is considered ^05?Yi7J6/ the latter Bb which falls on the prolongation of BQ is taken as negative. The products P,Aa^ and Q,Bb are called the virtual moments of P and Q. The virtual moments of P, Q^ and their resultant R (Fig. 42) are P.Ap^ Q-Aq^ R,Ar. Resuming the equations a = Q cos ^ + P COS a, = ^ sin ^ — P sin a, multiplying the first by cos f , the second by sin AP z z where a = CP. On the other hand in the triangle PAC we have AP' = s' = a*+r'- 2ar cos ^CD = a' + r* - 2ax; therefore m (a — x) dx ^"^ 2f 4/ (a- + r" -"2a^' ' or ^ m \^ adx xdx ) ^~2r I t/(a^-j-r«_"2^"" V (a^+T^-2axf \ * Now adx 1 J Via (a' + r' - 2axf V a" + r' — 2ax INFINITESIMAL CALCULUS. 25» and icdct V{a''\-r''-2uxf X Hence, substituting, and reducing, -4- ! • ^ a' 9 rri 2r ax — r + ^; a* V'a' + r' - 'lax and taking the integral from x^=^ — r to a; = r, and reducing, __ m j r {a - r) r {a -\-r) ) __ w . f a a-\-r that is, the total action is the same as if the whole mass of the shell were concentrated in C. The same being true of all the shells into which a solid sphere may be decomposed, it follows that the action of a sphere formed of homogene- ous shells is the same as if its mass were concentrated in its centre. Fig.J^ 109. If the point P were placed anywhere within the spherical shell (Pig. 45), the resultant of all the actions of the shell upon it would con- stantly be ==5 0, and therefore the point P wonld remain in equilibrium. For, in this case, the dif- ferential equation would be {x — a) dx cUp = m 2r i^{a' + r'-2axy 254 INFINITESIMAL 'CALCULUS. which differs only by tlie sign from the equation of the former case. Hence the integral will be m r* — ax , ^ and this, if taken from a? = — r to a? = r, will give _ m i r {r — a) r (r -{-a) ) __ ^ ~ 2ra* ( r —a r-^a f""' And this is true of all the shells whose radius is greater than CP. Accordingly, if the point P were placed within a solid sphere, it would be attracted as if the shells beyond CP had no existence ; that is, it would only feel the attraction of the nucleus PQH. Corollary. If an opening were made along one of the diameters of the earth, and a body allowed to fall through it, the body (abstraction being made from the resistance of the air) would be urged to- wards the centre by an action varying as the simple distance from the centre. For, let r be the radius, and p the density (sup- posed uniform) of the earth. Its mass will then be 47rr* -^ p\ whilst the mass of the nucleus PQR will be -^— />, s being its radius. Now, the action of the earth at its surface is ^ = -^ p —^ , and the action of the nucleus at a distance s from the centre is g^ = -^- p -^ ; whence 4 g' :g::s:r, or g' = g -• INFINITESIMAL CALCULUS. 255 Hence the equation for the movement of a point approaching the centre, is d^8 _ gs Multiplying by 2ds, and integrating, we have (§)■= - •? + « Making ^ = when s = Ty we have C =i — ^ Therefore (§)■= '•=?('■- *•)• (1) When the body reaches the centre, then 5 = 0^ and v=- Vgi\ which is the maxhnum velocity. When the body has reached the centre, its velocity will carry it further on, and, as s changes its sign, the motion will be retarded instead of accelerated, until V reduces to zero when ^= — r. Then the body will fall again towards the centre, and meas- ure backward the same diameter, and perform a continuous series of oscillations of the same kind. Extracting the root of equation (1) and taking the radical negatively, because ds and dt have op- posite signs, we find ds dt -^.- and this integrated from * — r to s=: — r, gives t = 7T A/-. y g This is the time of one entire excursion. This time 256 INFINITESIMAL CALCULUS. is equal to that in which a body would measure the semi-circumference rr witli a uniform velocity = VyT ; for, if t Vgr = ;rr, then ^ = ;r A/ - • - The time t of the excursion is independent of the distance from which the body begins to fall. For, since we have r:g::s:g\ we can replace the radical A/ - by 4/ -, , without altering the value of t. Hence all the excursions will be isochronous, whatever may be their iimpli- tude. But these results would be greatly modified by the resistance of the air, which we have neglected. Centre of Gravity, 110. The centre of gravity of a body is a point within the body, through which the resultant of the actions of gravity on each particle of the body always passes. All these actions are directed towards the centre of the earth ; yet they may, without error, be considered parallel. Hence their resultant is their sum (No. 105). It is obvious that the centre of gravity of a straight line is at its middle point ; also, that the centre of gravity of a plane figure is in that plane, and if the figure has a line of symmetry, its centre of gravity is on that line. In like manner, if a solid has a plane of symmetry, its centre of gravity is in that plane. The centre of gravity of a homogeneous body INFINITESIMAL CALCULUS. 257 does not depend on the intensity of gravity or ou the density of the body. Its position depends only on the form of its volume. We may therefore sub- stitute volumes for masses and weights, and con- sider only the relative position of the elements of which the body is com loosed. Let M (Fig. 46) be a homogeneous body of any form. Draw rectangular axes, and let the plane XY be horizontal. The action of gravity will be parallel to the axis AZ. Let m be an element of the body, and let its co-ordinates he x= qn^ y =pny z = mn. If dv be the volume of tht^ element ^ ^ m, its moment with re- spect to the axis AY will be xdv. Every other element of the body M will give a similar mo ment, the value of x varying between the limits of the body. Hence the sum of the moments of all the elements with respect to the axis A Fwill be / xdv. Ijet now be the centre of gravity of the body, and Xo = i>(7, y^ = BO^ z^ = 0(7, its co-ordinates. Since the resultant of the actions of gravity passes through 0, the moment of the resultant with re- spect to the axis ^ F will be a?b / dv. Hence, by the theory of moments (No. 106), Xq f do = /xdv; 258 INFINITESIMAL CALCULUS. and therefore ^ fxdo If the moments were taken with respect to the axis AX^ we would find in like manner ^0 = fdr. ' and if the figure were turned about so as to make the axis of x vertical, we would have also ;^o = f zdv 7^ Such are the values of the co-ordinates of the centre of gravity of the body. 111. Centre of gravity of a circular arc. Let the axis OX (Fig. 4T| bisect the arc ABC. Then OX will be a line of symmetry, and the centre of gravity will lie on OX. Let AC^=c be the chord, and OA = r the radius of the circular arc. If be the origin of co-ordinates, the equation of the o^ circle will be \ and therefore dv=VM+W=dyA/^=r^ = -^:^. " " \ 0^ X Vr'—y' / INFINITESIMAL CALCULUS. 25^ Substituting in the above expression for a^ we have frdy /rdy and integrating from y = — ic to y=i ^c^ re TO ( . . c .A c\\ arc.A^C* Making the angle AOX = ??, we have c = 2r sin t?^ whilst the arc ABC^ 2r& ; whence we get also sirt(? 112. Centre of gravity of a circular segment. Referring to Fig. 47, where the segment ABC is bisected by the axis 0X\ we find dv = 2ydx = ^^ ' - , xdv = 2y^dy: y r" — 2/' whence ^^- / - 2y^dy_ '^ J )/r^ ^^ and integrating from 2/ = to y=y^ by formula. (33), r^^= ^?^ r'sin-^^ — 2/|/r» — 2/« 2(50 INFINITESIMAL CALCULUS. and, since y =r sin cos J dmi.z cos OG^JS, But z cos 0(?-fi? = (7j£r= the lever arm of the mass dm with respect to the axis passing through the centre of gravity of the body. Hence / zdm cos OOE is the algebraic sum of the moments of all the par- ticles of the body with respect to the axis passing through its centre of gravity ; and this sum, by the ^ I 266 INFINITESIMAL CALCULUS. principle of moments (No. 106) must be =0, be- cause the moment of their resultant is also = 0. We have, therefore, simply mV = / fdm + mp*. Now / z^dm is the moment of inertia of the mass m with respect to the axis passing through the centre of gravity. Denoting it by w;Vj we shall have Therefore, tJie moment of inertia of a body with re- spect to any axis is equal to the moment qf inertia with respect to a parallel axis through the centre of gravity of the hody^ plus the mass of the body into the sqicare qf the distance between the ttoo a^es. 131. Moment of iner tia of a rectangle. Let PQ (Pig. 61) be an axis passing through the centre of gravity of a rectangle, and lying in the plane of the rectangle perjoendicularly to its length AB. Let m be the mass of the rectangle, and AB = 2a. Tlie iniinitesimal element CD = dm placed at any distance x from the axis, will be found by the proportion 2a : dx::m : dm; or dm=i— dx. Substituting in (1), we have Fig. 51 p G • • A qI D B INFINITESIMAL CALCULUS. 267 and integratmg from a? = — a to a? = a, and reduc- ing, m1^=zvi Hence the radius of gyration is here Jc = a V3 This result is independent of the altitude of the rectangle. Hence considering the straight line AB as a rectangle having an infinitesimal altitude, its a' moment of inertia will also he m -^ ^ m denoting the mass of the line. 122. Moment of inertia of a circle, when the axis coincides with a diameter AB, Let OC=r (Fig. 52) be the radius of the circle, CD = dm an element of its area parallel to the axis, 0^= x its distance from the centre O. We shall have TTT* : 2pdx ::m: dm^ or dm = — j ydx. Substituting in (1) and remembering that 2/ is = Vr' — x% we have By formula (33) we have f(r'-x^Yx'dx= ~|(r'-aj')* + ^y^C^"- a^)^dxj and by formula (34), 268 INFINITESIMAL CALCULUS. X r r{f^a?^dx = h i/7^^^ + 2 si^"' Substituting, and taking the integral from x to ir = r, we have 4 Hence the radius of gyration is here * = 2 • = — r Tig. 53 123. Moment of inertia of a circle^ when the axis through the centre is perpendictUar to tJie plane of the circle. With a radius 00 =x (Pig. 53) describe a circle, and give to its circum- ference a width dx. Then 27:a^dx will represent an element of the area, and thus ;rr* : 27rxdx::m : dm; or dm = — T" xdx. Substituting in (1), and integrating from rr = to a? = r, we have This formula is independent of the thickness of the circular plate ; hence it will be true for any thick- ness. It therefore expresses the moment of inertia of a solid cylinder of any length, revolving about its axis. INFINITESIMAL CALCULUS. 269 124. Moment of inertia of a circular ring with respect to an axis perpendicular to its plane. Let r and r' be the extreme radii of the ring, and m its mass. Taking x between r and r', we have ;r(r* — r") : 27rxdx::m : dm; and dm = -5 75 a?e?a;. Substituting in (1), and integrating from x=:r^ to x = r, we find ,, * m r* — r'* r* + r'* mk* = -^ -5 7i = m — ^ 2 r* — r ' 2 This formula, being independent of the thickness of the ring, will be true for a hollow cylinder of any length. 135. Moment of inertia of a cylinder with re- spect to an axis perpendicular to the a^is of the cylinder. Let the axis PQ (Pig. 54) be taken tlirough the centre of gravity of the cylinder, and let BD be an element perpendicular to the axis of the cylinder, at a distance 0(7= x from the axis of rotation. Let r be the radius of the cylinder, and // the mass of the element BD. The moment of inertia of this element with respect to one of its P B diameters would be (No. 122) = // j ; but its mo- ment of inertia with respect to the axis PQ parallel to that diameter, and placed at a distance OC=zx^ will be /^ ( J- + ^') 5 as we have shown (No. 120). 270 INFIXITESIMAL CALCULUS. Now, let 2a be the length, and m the mass, of the cylinder. We shall have /jl: 7n::dx :2ay or /i = ^ dx; and therefore the moment of inertia of the whole cylinder will be found by integmting the expres- sion X- I J- +^j ^ between x= -^ a and ic = a. Integrating, and reducing, we shall find mk' = m (^ + 1). 136. Moment of inertia of a sphere. Let the axis pass through the centre of the sphere. Let r be the radius of the sphere and fi the mass of an elementary segment perpendicular to the * axis, placed at a distance x from the centre, and having a radius y= Vr^ -- x\ The moment of inertia of this element with respect to the axis is (No. 123) /i| , or I (r'-rr>). Now, if m be the mass of the sphere, we have fi:m:\ izy ax : -g- ; whence and substituting this value of // in the above ex- pression, we shall have for the moment of inertia b INFINITESIMAL CALCULUS. of the elementai'y segment Integrating this from a: =: — r fo x=^ri we get, after reduction, for the moment of inertia of the whole sphere. 137. Problem. As an application of the theory of the moments of inertia, let R and r (Pig. 55) be the radii of two solid cylinders having the same hori- zontal axis, M being the mass of the larger and m that of the smaller ; and let tlie weights p and P be applied to them respectively by a thread wrapped on their surfaces. If the two cylinders are so connected that they must rotate together, and if the weighty acts in a direction opposite to that of P, what will be the movement of these weights after a time fi Solution. Adapting to our case the formula (No. 119) ^ - _^_ '^ ~ f^dm' and reflecting that the moments of inertia of the 272 INFINITESIMAL CALCULUS. two solid cylinders are M -^ and m ^ respect- ively, and that the moments of the accelerating fonses are Pr and pB^ we shall have tP^_ Pr-Rp ,, Pr-Bp 2 2 Let now M' and m' be the masses of the weights P and p respectively. Then g being the action of gravity^ we have P = M'g, p = m'g. And there- fore dC ~^^ Mie-mr" ' and integrating from < = to ^ = ^, d» _ M't - Rm' « _ ^ 3f V - Rm' dt ""^^^ Mie-mr' ' ^-^^ MR-mr' ' Ourvilinear Movement. 188. A free point M (Fig. 56) cannot move in a curve, unless its direc- tion be continually ng.se changed by an action pro- ceeding from some other direction. Let us con- ceive this action decom- posed into two, X and Ty respectively parallel to the co-ordinate axes OX and Y. Draw MJN' normal, and JfT^ tangent o to the curve at the point Jf. Resolve X into MT INFINITESIMAL CALCULUS. 273 tangential, and Mp normal, also Y into MT^ tan- gential, and Mq normal to the curve. Then the forces X and Y may be replaced by a tangential force T=^ MT — MT\ and by a normal^ or centri- petal force N = J/p + Mq. Now, calling (? the angle that the element ds of the curve at M makes with the axis of abscissas, we have MT:= Xcos », MT = r sin *, Mp=:X sin &j Mq = Y cos ^. Hence T=Xcos»-Ysm^, i\r=Xsini? + rcos(?. But, according to our usual notation, dt' ' d^' . t, dy o dx , sm I? = :^ , cos ^ = ^y- ; ds ds Therefore dP ds df ds ' The first of these equations can be reduced to ^_ d {da? + dy') _ d {ds') _ 2dsd's _ d's ~ dt' .2ds "^ d^t' . 2ds "" ^tZ^d'/' eZ^' * The second, being multiplied and divided by ds\ becomes 274 INFINITESIMAL CALCULUS. ds* d^ydx — d^xdy . and thi8, according to the remark made by us on the expression for the i-adius of curvature when t is the independent variable (No. 46), will become Such is the expression of the centripetal accelera- tion. Hence the centripetal force is equal to the product of the mass into the square of the tangent tial velocity divided by the radius of curvature The moving point, while obeying the centripetal action, always keeps its tendency to follow a straight line, that is, the tangential direction, and thus to recede from the centre of curvature. This centrifugal tendency, so far as counteracted and thwarted by the centripetal action, is usually called the centrifugal for ce^ and its intensity is measured by that of the action by which it is thwarted. Hence the centrifugal force is equal to the centri- petal, and directly opposed to it. The centripetal and centrifugal forces are commonly called central forces. 139. Let a point M roll down a curve OMC (Pig. 57) under the action of gravity. Let OX and Z be the co-ordinate axes, and let the ordinates downward be positive. When the moving point has reached any position M, the action of gravity MOj by which its movement is accelerated, may be decomposed into MJV normal, and MT tangential to the curve. The first will be destroyed by the Infinitesimal calculus. 275 Fig, 57 resistance of the curve, whicli we assume to be in- variably fixed, and wliicli is thus playing the part of a centripetal force. The sec- ^ ond will have its whole effect. Let d^ be the angle that the curve at M makea with the \ axis OX Then MT=g^m»^g^, or d^s _ dy dP~^ di' Hence ^ds d^ dt dt = 2gdy, and (J^ = 2gy. This velocity is the velocity due to the height y (No. 100). Therefore the velocity acquired by a body rolling down a curve under the action of gravity, is equal to that which it acquires by falling freely through the same vertical height. This result is true not only when g is constant, but also when g varies according to a fixed law; for, even in this case, g may be regarded as con- stunt from element to element, inasmuch as the same law of variation applies to the elements of the curve and to those of rlie vertical line. Hence a body falling toward the suu on a spiral line will have the same final velocity as though it had fallen directly towards its centre. 130: The simple pendulum. A material point 276 INFINITSaiMAL CALCULUS: A (Fig. 08) suspended from a horizontal axis by a rigid line AO without weight, and free to oscil- late about that axis, con- stitutes a simple pendu- lum. Let AO = ?, the angle AOC = a, and the angle MOC = *. When the point A under the action of gravity reaches the point Jf, it will have acquired a ds velocity ^ = V^^ (No. 129), y being = DE. But d8 = ld^^ and y — OE— OB = I (cos * — cos a) ; therefore Z ^ = ^2gl (cos & — cos a) ; whence dt -yw d& 2^7 i^cos^ — cosa From Maclaurin's formula we have cos t/ = 1 — TT + 2 * 2.3.4 cosa=l-|' + 2;|^- • • • • « hence, if the arc J. (7 be small enough to allow us to neglect all the terms of the series after the sec- ond, we shall have cos t? — cos a:=.\((;f — #•), and INFINITESIMAL CALCULUS. 277 dt ~ y g' V^r^&' ' Integrating from «>— — afcot? = a, we shall have t -Vj-- This is the time of one excursion, or semi- oscilla- tion, of a simple pendulum, when the amplitude of the excursion is small; that is, not exceeding 9 or 10 degrees. 131. Planetary orbits. By the first of Kepler's laws, the orbits of planets are ellipses, of which one focus is occupied by the sun. Fig.J9 Let PMA (Fig. 69) be a - ^^ planetary orbit, and JF the /'''^ I^^M focus occupied by the sun. Let the planet, at a given PI instant dt, be moving from M towards the perihelion P. Make FM=:p^ the angle MFA=v, OQ = x, MQ^y, OF^c, and let

' = J yp\ whence This value of V substituted in (4) will lead to y-' = 2<^ 4 • -rr- . (5) But, if a and b be the major and minor semi-axes of the orbit, we have />' + />" = 2a, />>" = (a - c) (a + c) = a' - c* = b\ Substituting in (6), F" = 4--. (6) ^ I 280 INFiyiTESIMAL CALCULUS. Comparing equations (3) and (6), we obtain (1)* = ^^ l^"W7")+^~7']' which, owing to (6), will be reduced to And now, from the general equation ds' =ipdvy + dp* we have (i)"=(i')'+(iy <«> from which we shall easily eliminate ^ by con- sidering that, since the area described by the radius vector in the time dt is ^p^dv, the area de- I dfi scribed in the unit of time will be i/>' -37 . But we have already found that this area can be expressed hjiVy. Hence ^ dt ^ P' P dt- p ' \dt) - p' ' or substituting the value of V from equation (6), and reducing, /pdm\* _' ' whence y 9' V2ap-p*-h'' or, since 6' = a' — c*, _ _ fa ( adp {p-a)dp 1 y ^ ( V'c' _ (^ _ a)« "^ |/o' - (/> - a)" f ' of which the integral is If we reckon the time from the aphelion J., we shall have ^ = when /^rra + c? tliat is, when p — a = c; and then C= 0. Therefore .=/«(„ CO,-. e=£ +.^73^). „) Such is the expression for the time taken by the planet in measuring any portion AM of its orbit, p being the radius vector of the point M. 282 INFINITESIMAL CALCULUS. The time T employed in measuring the semi orbit AM'P will be found by taking pz=.a — c^ or /o — a = — c. Then (9) gives = a;r|/|. (10) This equation shows that the squares of the times of the revolutions of two planets are to each other directly as the cubes of the transverse axes of their orbits. This is the third among Kepler's laws. The same equation shows also that the time of a revolution is independent of the minor axis h of the orbit. Assume ft so small, that the ellipse may sensibly be reduced to a double straight line. Then the focus will sensiblv coincide with the ex- tremity F of the transverse axis, and T would be the time taken by the planet in falling directly upon the sun from the distance AP = 2a. Prom (10) we have 4/ - = — . be transformed into ( ;r c an y Hence (9) may W)\' and if e be the eccentricity of the orbit, we may substitute e f or - , and reduce the equation to the a form t=T l^ 1 - cos 7Z -1 vl+.-v-l"-^. I ^ INFINITESIMAL CALCULUS. 283 Lastly, making e the equation mil take the form ^ -=zCOSI> (11) t^T{^-±^). (12) The formulas (11) and (12), and the polar equa- tion of the ellipse, which is g = , ^-^ (13) a 1 — e cos V ^ ^ will suffice to determine the time taken by the planet in measuring any given angle v reckoned from the aphelion, when the eccentricity of the orbit is known. Scholium. The eccentricity of the terrestrial orbit being e = 0.016833, and the earth, during the tropical year, measuring only 359"^ 59' 9''.8 around the sun, the longitude of the aphelion (which is reckoned from the vernal equinox, and which on the 1st of January, 1800, was 99° 30' 8^39) is in- creasing every year by SO''. 2. The length of the tropical year is = 365''.242256. Will the student, with these data, and with the aid of the last three formulas, try to determine the length of the four seasons for some given year 3